iCAL Modern Calculus
with R and Python
Version 1.0 developed from May 2021 for Cal III by
Dr. Samuel Shen, Distinguished Professor
San Diego State
University, California, USA
https://shen.sdsu.edu
Email: sshen@sdsu.edu
Chapter 10:Stokes-Cartan Theorem
setwd('/Users/sshen/CalculusR')Example of computing the area of an ellipse by a planimeter
n=1000
a=4
b=2
t=seq(0, 2*pi, length = n+1)
x=a*cos(t)
y=b*sin(t)
s=rep(0,n)
for (i in 1:n) {
s[i] = (1/2)*(-y[i]*(x[i+1]-x[i]) + x[i]*(y[i+1]-y[i]))
}
A=sum(s)
A## [1] 25.13258A simplified version for the area of an ellipse by a planimeter
n=1000
a=4
b=2
t=seq(0, 2*pi, length = n+1)
x=a*cos(t)
x## [1] 4.000000e+00 3.999921e+00 3.999684e+00 3.999289e+00 3.998737e+00
## [6] 3.998026e+00 3.997158e+00 3.996132e+00 3.994948e+00 3.993606e+00
## [11] 3.992107e+00 3.990450e+00 3.988636e+00 3.986664e+00 3.984534e+00
## [16] 3.982248e+00 3.979804e+00 3.977203e+00 3.974445e+00 3.971530e+00
## [21] 3.968459e+00 3.965231e+00 3.961846e+00 3.958304e+00 3.954607e+00
## [26] 3.950753e+00 3.946744e+00 3.942578e+00 3.938257e+00 3.933781e+00
## [31] 3.929149e+00 3.924362e+00 3.919420e+00 3.914324e+00 3.909072e+00
## [36] 3.903667e+00 3.898107e+00 3.892394e+00 3.886527e+00 3.880506e+00
## [41] 3.874333e+00 3.868006e+00 3.861527e+00 3.854895e+00 3.848111e+00
## [46] 3.841175e+00 3.834087e+00 3.826848e+00 3.819458e+00 3.811917e+00
## [51] 3.804226e+00 3.796385e+00 3.788393e+00 3.780252e+00 3.771962e+00
## [56] 3.763523e+00 3.754935e+00 3.746200e+00 3.737316e+00 3.728284e+00
## [61] 3.719106e+00 3.709781e+00 3.700309e+00 3.690691e+00 3.680927e+00
## [66] 3.671019e+00 3.660965e+00 3.650766e+00 3.640424e+00 3.629938e+00
## [71] 3.619308e+00 3.608536e+00 3.597621e+00 3.586564e+00 3.575366e+00
## [76] 3.564026e+00 3.552546e+00 3.540925e+00 3.529165e+00 3.517265e+00
## [81] 3.505227e+00 3.493050e+00 3.480735e+00 3.468283e+00 3.455694e+00
## [86] 3.442968e+00 3.430107e+00 3.417110e+00 3.403978e+00 3.390712e+00
## [91] 3.377312e+00 3.363778e+00 3.350112e+00 3.336314e+00 3.322384e+00
## [96] 3.308322e+00 3.294130e+00 3.279808e+00 3.265357e+00 3.250777e+00
## [101] 3.236068e+00 3.221232e+00 3.206268e+00 3.191178e+00 3.175962e+00
## [106] 3.160620e+00 3.145154e+00 3.129563e+00 3.113849e+00 3.098012e+00
## [111] 3.082053e+00 3.065972e+00 3.049770e+00 3.033448e+00 3.017006e+00
## [116] 3.000444e+00 2.983765e+00 2.966967e+00 2.950052e+00 2.933021e+00
## [121] 2.915875e+00 2.898613e+00 2.881236e+00 2.863746e+00 2.846143e+00
## [126] 2.828427e+00 2.810600e+00 2.792662e+00 2.774613e+00 2.756455e+00
## [131] 2.738188e+00 2.719814e+00 2.701331e+00 2.682742e+00 2.664047e+00
## [136] 2.645247e+00 2.626343e+00 2.607335e+00 2.588224e+00 2.569011e+00
## [141] 2.549696e+00 2.530281e+00 2.510765e+00 2.491151e+00 2.471438e+00
## [146] 2.451628e+00 2.431721e+00 2.411718e+00 2.391620e+00 2.371427e+00
## [151] 2.351141e+00 2.330762e+00 2.310291e+00 2.289729e+00 2.269076e+00
## [156] 2.248334e+00 2.227502e+00 2.206583e+00 2.185577e+00 2.164485e+00
## [161] 2.143307e+00 2.122045e+00 2.100699e+00 2.079269e+00 2.057758e+00
## [166] 2.036166e+00 2.014493e+00 1.992740e+00 1.970909e+00 1.949001e+00
## [171] 1.927015e+00 1.904953e+00 1.882816e+00 1.860604e+00 1.838319e+00
## [176] 1.815962e+00 1.793533e+00 1.771033e+00 1.748463e+00 1.725824e+00
## [181] 1.703117e+00 1.680343e+00 1.657502e+00 1.634596e+00 1.611626e+00
## [186] 1.588592e+00 1.565495e+00 1.542336e+00 1.519116e+00 1.495837e+00
## [191] 1.472498e+00 1.449101e+00 1.425648e+00 1.402137e+00 1.378572e+00
## [196] 1.354952e+00 1.331278e+00 1.307552e+00 1.283774e+00 1.259946e+00
## [201] 1.236068e+00 1.212141e+00 1.188166e+00 1.164145e+00 1.140077e+00
## [206] 1.115964e+00 1.091808e+00 1.067608e+00 1.043366e+00 1.019083e+00
## [211] 9.947595e-01 9.703969e-01 9.459960e-01 9.215577e-01 8.970830e-01
## [216] 8.725730e-01 8.480284e-01 8.234504e-01 7.988399e-01 7.741979e-01
## [221] 7.495253e-01 7.248231e-01 7.000922e-01 6.753338e-01 6.505487e-01
## [226] 6.257379e-01 6.009024e-01 5.760431e-01 5.511612e-01 5.262574e-01
## [231] 5.013329e-01 4.763886e-01 4.514255e-01 4.264446e-01 4.014469e-01
## [236] 3.764333e-01 3.514048e-01 3.263624e-01 3.013072e-01 2.762401e-01
## [241] 2.511621e-01 2.260741e-01 2.009773e-01 1.758725e-01 1.507607e-01
## [246] 1.256430e-01 1.005204e-01 7.539376e-02 5.026416e-02 2.513258e-02
## [251] 2.449294e-16 -2.513258e-02 -5.026416e-02 -7.539376e-02 -1.005204e-01
## [256] -1.256430e-01 -1.507607e-01 -1.758725e-01 -2.009773e-01 -2.260741e-01
## [261] -2.511621e-01 -2.762401e-01 -3.013072e-01 -3.263624e-01 -3.514048e-01
## [266] -3.764333e-01 -4.014469e-01 -4.264446e-01 -4.514255e-01 -4.763886e-01
## [271] -5.013329e-01 -5.262574e-01 -5.511612e-01 -5.760431e-01 -6.009024e-01
## [276] -6.257379e-01 -6.505487e-01 -6.753338e-01 -7.000922e-01 -7.248231e-01
## [281] -7.495253e-01 -7.741979e-01 -7.988399e-01 -8.234504e-01 -8.480284e-01
## [286] -8.725730e-01 -8.970830e-01 -9.215577e-01 -9.459960e-01 -9.703969e-01
## [291] -9.947595e-01 -1.019083e+00 -1.043366e+00 -1.067608e+00 -1.091808e+00
## [296] -1.115964e+00 -1.140077e+00 -1.164145e+00 -1.188166e+00 -1.212141e+00
## [301] -1.236068e+00 -1.259946e+00 -1.283774e+00 -1.307552e+00 -1.331278e+00
## [306] -1.354952e+00 -1.378572e+00 -1.402137e+00 -1.425648e+00 -1.449101e+00
## [311] -1.472498e+00 -1.495837e+00 -1.519116e+00 -1.542336e+00 -1.565495e+00
## [316] -1.588592e+00 -1.611626e+00 -1.634596e+00 -1.657502e+00 -1.680343e+00
## [321] -1.703117e+00 -1.725824e+00 -1.748463e+00 -1.771033e+00 -1.793533e+00
## [326] -1.815962e+00 -1.838319e+00 -1.860604e+00 -1.882816e+00 -1.904953e+00
## [331] -1.927015e+00 -1.949001e+00 -1.970909e+00 -1.992740e+00 -2.014493e+00
## [336] -2.036166e+00 -2.057758e+00 -2.079269e+00 -2.100699e+00 -2.122045e+00
## [341] -2.143307e+00 -2.164485e+00 -2.185577e+00 -2.206583e+00 -2.227502e+00
## [346] -2.248334e+00 -2.269076e+00 -2.289729e+00 -2.310291e+00 -2.330762e+00
## [351] -2.351141e+00 -2.371427e+00 -2.391620e+00 -2.411718e+00 -2.431721e+00
## [356] -2.451628e+00 -2.471438e+00 -2.491151e+00 -2.510765e+00 -2.530281e+00
## [361] -2.549696e+00 -2.569011e+00 -2.588224e+00 -2.607335e+00 -2.626343e+00
## [366] -2.645247e+00 -2.664047e+00 -2.682742e+00 -2.701331e+00 -2.719814e+00
## [371] -2.738188e+00 -2.756455e+00 -2.774613e+00 -2.792662e+00 -2.810600e+00
## [376] -2.828427e+00 -2.846143e+00 -2.863746e+00 -2.881236e+00 -2.898613e+00
## [381] -2.915875e+00 -2.933021e+00 -2.950052e+00 -2.966967e+00 -2.983765e+00
## [386] -3.000444e+00 -3.017006e+00 -3.033448e+00 -3.049770e+00 -3.065972e+00
## [391] -3.082053e+00 -3.098012e+00 -3.113849e+00 -3.129563e+00 -3.145154e+00
## [396] -3.160620e+00 -3.175962e+00 -3.191178e+00 -3.206268e+00 -3.221232e+00
## [401] -3.236068e+00 -3.250777e+00 -3.265357e+00 -3.279808e+00 -3.294130e+00
## [406] -3.308322e+00 -3.322384e+00 -3.336314e+00 -3.350112e+00 -3.363778e+00
## [411] -3.377312e+00 -3.390712e+00 -3.403978e+00 -3.417110e+00 -3.430107e+00
## [416] -3.442968e+00 -3.455694e+00 -3.468283e+00 -3.480735e+00 -3.493050e+00
## [421] -3.505227e+00 -3.517265e+00 -3.529165e+00 -3.540925e+00 -3.552546e+00
## [426] -3.564026e+00 -3.575366e+00 -3.586564e+00 -3.597621e+00 -3.608536e+00
## [431] -3.619308e+00 -3.629938e+00 -3.640424e+00 -3.650766e+00 -3.660965e+00
## [436] -3.671019e+00 -3.680927e+00 -3.690691e+00 -3.700309e+00 -3.709781e+00
## [441] -3.719106e+00 -3.728284e+00 -3.737316e+00 -3.746200e+00 -3.754935e+00
## [446] -3.763523e+00 -3.771962e+00 -3.780252e+00 -3.788393e+00 -3.796385e+00
## [451] -3.804226e+00 -3.811917e+00 -3.819458e+00 -3.826848e+00 -3.834087e+00
## [456] -3.841175e+00 -3.848111e+00 -3.854895e+00 -3.861527e+00 -3.868006e+00
## [461] -3.874333e+00 -3.880506e+00 -3.886527e+00 -3.892394e+00 -3.898107e+00
## [466] -3.903667e+00 -3.909072e+00 -3.914324e+00 -3.919420e+00 -3.924362e+00
## [471] -3.929149e+00 -3.933781e+00 -3.938257e+00 -3.942578e+00 -3.946744e+00
## [476] -3.950753e+00 -3.954607e+00 -3.958304e+00 -3.961846e+00 -3.965231e+00
## [481] -3.968459e+00 -3.971530e+00 -3.974445e+00 -3.977203e+00 -3.979804e+00
## [486] -3.982248e+00 -3.984534e+00 -3.986664e+00 -3.988636e+00 -3.990450e+00
## [491] -3.992107e+00 -3.993606e+00 -3.994948e+00 -3.996132e+00 -3.997158e+00
## [496] -3.998026e+00 -3.998737e+00 -3.999289e+00 -3.999684e+00 -3.999921e+00
## [501] -4.000000e+00 -3.999921e+00 -3.999684e+00 -3.999289e+00 -3.998737e+00
## [506] -3.998026e+00 -3.997158e+00 -3.996132e+00 -3.994948e+00 -3.993606e+00
## [511] -3.992107e+00 -3.990450e+00 -3.988636e+00 -3.986664e+00 -3.984534e+00
## [516] -3.982248e+00 -3.979804e+00 -3.977203e+00 -3.974445e+00 -3.971530e+00
## [521] -3.968459e+00 -3.965231e+00 -3.961846e+00 -3.958304e+00 -3.954607e+00
## [526] -3.950753e+00 -3.946744e+00 -3.942578e+00 -3.938257e+00 -3.933781e+00
## [531] -3.929149e+00 -3.924362e+00 -3.919420e+00 -3.914324e+00 -3.909072e+00
## [536] -3.903667e+00 -3.898107e+00 -3.892394e+00 -3.886527e+00 -3.880506e+00
## [541] -3.874333e+00 -3.868006e+00 -3.861527e+00 -3.854895e+00 -3.848111e+00
## [546] -3.841175e+00 -3.834087e+00 -3.826848e+00 -3.819458e+00 -3.811917e+00
## [551] -3.804226e+00 -3.796385e+00 -3.788393e+00 -3.780252e+00 -3.771962e+00
## [556] -3.763523e+00 -3.754935e+00 -3.746200e+00 -3.737316e+00 -3.728284e+00
## [561] -3.719106e+00 -3.709781e+00 -3.700309e+00 -3.690691e+00 -3.680927e+00
## [566] -3.671019e+00 -3.660965e+00 -3.650766e+00 -3.640424e+00 -3.629938e+00
## [571] -3.619308e+00 -3.608536e+00 -3.597621e+00 -3.586564e+00 -3.575366e+00
## [576] -3.564026e+00 -3.552546e+00 -3.540925e+00 -3.529165e+00 -3.517265e+00
## [581] -3.505227e+00 -3.493050e+00 -3.480735e+00 -3.468283e+00 -3.455694e+00
## [586] -3.442968e+00 -3.430107e+00 -3.417110e+00 -3.403978e+00 -3.390712e+00
## [591] -3.377312e+00 -3.363778e+00 -3.350112e+00 -3.336314e+00 -3.322384e+00
## [596] -3.308322e+00 -3.294130e+00 -3.279808e+00 -3.265357e+00 -3.250777e+00
## [601] -3.236068e+00 -3.221232e+00 -3.206268e+00 -3.191178e+00 -3.175962e+00
## [606] -3.160620e+00 -3.145154e+00 -3.129563e+00 -3.113849e+00 -3.098012e+00
## [611] -3.082053e+00 -3.065972e+00 -3.049770e+00 -3.033448e+00 -3.017006e+00
## [616] -3.000444e+00 -2.983765e+00 -2.966967e+00 -2.950052e+00 -2.933021e+00
## [621] -2.915875e+00 -2.898613e+00 -2.881236e+00 -2.863746e+00 -2.846143e+00
## [626] -2.828427e+00 -2.810600e+00 -2.792662e+00 -2.774613e+00 -2.756455e+00
## [631] -2.738188e+00 -2.719814e+00 -2.701331e+00 -2.682742e+00 -2.664047e+00
## [636] -2.645247e+00 -2.626343e+00 -2.607335e+00 -2.588224e+00 -2.569011e+00
## [641] -2.549696e+00 -2.530281e+00 -2.510765e+00 -2.491151e+00 -2.471438e+00
## [646] -2.451628e+00 -2.431721e+00 -2.411718e+00 -2.391620e+00 -2.371427e+00
## [651] -2.351141e+00 -2.330762e+00 -2.310291e+00 -2.289729e+00 -2.269076e+00
## [656] -2.248334e+00 -2.227502e+00 -2.206583e+00 -2.185577e+00 -2.164485e+00
## [661] -2.143307e+00 -2.122045e+00 -2.100699e+00 -2.079269e+00 -2.057758e+00
## [666] -2.036166e+00 -2.014493e+00 -1.992740e+00 -1.970909e+00 -1.949001e+00
## [671] -1.927015e+00 -1.904953e+00 -1.882816e+00 -1.860604e+00 -1.838319e+00
## [676] -1.815962e+00 -1.793533e+00 -1.771033e+00 -1.748463e+00 -1.725824e+00
## [681] -1.703117e+00 -1.680343e+00 -1.657502e+00 -1.634596e+00 -1.611626e+00
## [686] -1.588592e+00 -1.565495e+00 -1.542336e+00 -1.519116e+00 -1.495837e+00
## [691] -1.472498e+00 -1.449101e+00 -1.425648e+00 -1.402137e+00 -1.378572e+00
## [696] -1.354952e+00 -1.331278e+00 -1.307552e+00 -1.283774e+00 -1.259946e+00
## [701] -1.236068e+00 -1.212141e+00 -1.188166e+00 -1.164145e+00 -1.140077e+00
## [706] -1.115964e+00 -1.091808e+00 -1.067608e+00 -1.043366e+00 -1.019083e+00
## [711] -9.947595e-01 -9.703969e-01 -9.459960e-01 -9.215577e-01 -8.970830e-01
## [716] -8.725730e-01 -8.480284e-01 -8.234504e-01 -7.988399e-01 -7.741979e-01
## [721] -7.495253e-01 -7.248231e-01 -7.000922e-01 -6.753338e-01 -6.505487e-01
## [726] -6.257379e-01 -6.009024e-01 -5.760431e-01 -5.511612e-01 -5.262574e-01
## [731] -5.013329e-01 -4.763886e-01 -4.514255e-01 -4.264446e-01 -4.014469e-01
## [736] -3.764333e-01 -3.514048e-01 -3.263624e-01 -3.013072e-01 -2.762401e-01
## [741] -2.511621e-01 -2.260741e-01 -2.009773e-01 -1.758725e-01 -1.507607e-01
## [746] -1.256430e-01 -1.005204e-01 -7.539376e-02 -5.026416e-02 -2.513258e-02
## [751] -7.347881e-16 2.513258e-02 5.026416e-02 7.539376e-02 1.005204e-01
## [756] 1.256430e-01 1.507607e-01 1.758725e-01 2.009773e-01 2.260741e-01
## [761] 2.511621e-01 2.762401e-01 3.013072e-01 3.263624e-01 3.514048e-01
## [766] 3.764333e-01 4.014469e-01 4.264446e-01 4.514255e-01 4.763886e-01
## [771] 5.013329e-01 5.262574e-01 5.511612e-01 5.760431e-01 6.009024e-01
## [776] 6.257379e-01 6.505487e-01 6.753338e-01 7.000922e-01 7.248231e-01
## [781] 7.495253e-01 7.741979e-01 7.988399e-01 8.234504e-01 8.480284e-01
## [786] 8.725730e-01 8.970830e-01 9.215577e-01 9.459960e-01 9.703969e-01
## [791] 9.947595e-01 1.019083e+00 1.043366e+00 1.067608e+00 1.091808e+00
## [796] 1.115964e+00 1.140077e+00 1.164145e+00 1.188166e+00 1.212141e+00
## [801] 1.236068e+00 1.259946e+00 1.283774e+00 1.307552e+00 1.331278e+00
## [806] 1.354952e+00 1.378572e+00 1.402137e+00 1.425648e+00 1.449101e+00
## [811] 1.472498e+00 1.495837e+00 1.519116e+00 1.542336e+00 1.565495e+00
## [816] 1.588592e+00 1.611626e+00 1.634596e+00 1.657502e+00 1.680343e+00
## [821] 1.703117e+00 1.725824e+00 1.748463e+00 1.771033e+00 1.793533e+00
## [826] 1.815962e+00 1.838319e+00 1.860604e+00 1.882816e+00 1.904953e+00
## [831] 1.927015e+00 1.949001e+00 1.970909e+00 1.992740e+00 2.014493e+00
## [836] 2.036166e+00 2.057758e+00 2.079269e+00 2.100699e+00 2.122045e+00
## [841] 2.143307e+00 2.164485e+00 2.185577e+00 2.206583e+00 2.227502e+00
## [846] 2.248334e+00 2.269076e+00 2.289729e+00 2.310291e+00 2.330762e+00
## [851] 2.351141e+00 2.371427e+00 2.391620e+00 2.411718e+00 2.431721e+00
## [856] 2.451628e+00 2.471438e+00 2.491151e+00 2.510765e+00 2.530281e+00
## [861] 2.549696e+00 2.569011e+00 2.588224e+00 2.607335e+00 2.626343e+00
## [866] 2.645247e+00 2.664047e+00 2.682742e+00 2.701331e+00 2.719814e+00
## [871] 2.738188e+00 2.756455e+00 2.774613e+00 2.792662e+00 2.810600e+00
## [876] 2.828427e+00 2.846143e+00 2.863746e+00 2.881236e+00 2.898613e+00
## [881] 2.915875e+00 2.933021e+00 2.950052e+00 2.966967e+00 2.983765e+00
## [886] 3.000444e+00 3.017006e+00 3.033448e+00 3.049770e+00 3.065972e+00
## [891] 3.082053e+00 3.098012e+00 3.113849e+00 3.129563e+00 3.145154e+00
## [896] 3.160620e+00 3.175962e+00 3.191178e+00 3.206268e+00 3.221232e+00
## [901] 3.236068e+00 3.250777e+00 3.265357e+00 3.279808e+00 3.294130e+00
## [906] 3.308322e+00 3.322384e+00 3.336314e+00 3.350112e+00 3.363778e+00
## [911] 3.377312e+00 3.390712e+00 3.403978e+00 3.417110e+00 3.430107e+00
## [916] 3.442968e+00 3.455694e+00 3.468283e+00 3.480735e+00 3.493050e+00
## [921] 3.505227e+00 3.517265e+00 3.529165e+00 3.540925e+00 3.552546e+00
## [926] 3.564026e+00 3.575366e+00 3.586564e+00 3.597621e+00 3.608536e+00
## [931] 3.619308e+00 3.629938e+00 3.640424e+00 3.650766e+00 3.660965e+00
## [936] 3.671019e+00 3.680927e+00 3.690691e+00 3.700309e+00 3.709781e+00
## [941] 3.719106e+00 3.728284e+00 3.737316e+00 3.746200e+00 3.754935e+00
## [946] 3.763523e+00 3.771962e+00 3.780252e+00 3.788393e+00 3.796385e+00
## [951] 3.804226e+00 3.811917e+00 3.819458e+00 3.826848e+00 3.834087e+00
## [956] 3.841175e+00 3.848111e+00 3.854895e+00 3.861527e+00 3.868006e+00
## [961] 3.874333e+00 3.880506e+00 3.886527e+00 3.892394e+00 3.898107e+00
## [966] 3.903667e+00 3.909072e+00 3.914324e+00 3.919420e+00 3.924362e+00
## [971] 3.929149e+00 3.933781e+00 3.938257e+00 3.942578e+00 3.946744e+00
## [976] 3.950753e+00 3.954607e+00 3.958304e+00 3.961846e+00 3.965231e+00
## [981] 3.968459e+00 3.971530e+00 3.974445e+00 3.977203e+00 3.979804e+00
## [986] 3.982248e+00 3.984534e+00 3.986664e+00 3.988636e+00 3.990450e+00
## [991] 3.992107e+00 3.993606e+00 3.994948e+00 3.996132e+00 3.997158e+00
## [996] 3.998026e+00 3.998737e+00 3.999289e+00 3.999684e+00 3.999921e+00
## [1001] 4.000000e+00y=b*sin(t)
y## [1] 0.000000e+00 1.256629e-02 2.513208e-02 3.769688e-02 5.026019e-02
## [6] 6.282152e-02 7.538037e-02 8.793624e-02 1.004886e-01 1.130371e-01
## [11] 1.255810e-01 1.381201e-01 1.506536e-01 1.631812e-01 1.757024e-01
## [16] 1.882166e-01 2.007234e-01 2.132223e-01 2.257128e-01 2.381943e-01
## [21] 2.506665e-01 2.631287e-01 2.755806e-01 2.880216e-01 3.004512e-01
## [26] 3.128689e-01 3.252743e-01 3.376669e-01 3.500461e-01 3.624115e-01
## [31] 3.747626e-01 3.870989e-01 3.994200e-01 4.117252e-01 4.240142e-01
## [36] 4.362865e-01 4.485415e-01 4.607789e-01 4.729980e-01 4.851985e-01
## [41] 4.973798e-01 5.095415e-01 5.216830e-01 5.338040e-01 5.459039e-01
## [46] 5.579822e-01 5.700385e-01 5.820723e-01 5.940832e-01 6.060705e-01
## [51] 6.180340e-01 6.299730e-01 6.418872e-01 6.537761e-01 6.656391e-01
## [56] 6.774758e-01 6.892858e-01 7.010686e-01 7.128238e-01 7.245507e-01
## [61] 7.362491e-01 7.479184e-01 7.595582e-01 7.711680e-01 7.827473e-01
## [66] 7.942958e-01 8.058129e-01 8.172981e-01 8.287512e-01 8.401715e-01
## [71] 8.515586e-01 8.629121e-01 8.742315e-01 8.855165e-01 8.967664e-01
## [76] 9.079810e-01 9.191597e-01 9.303022e-01 9.414079e-01 9.524764e-01
## [81] 9.635073e-01 9.745003e-01 9.854547e-01 9.963702e-01 1.007246e+00
## [86] 1.018083e+00 1.028879e+00 1.039635e+00 1.050349e+00 1.061022e+00
## [91] 1.071654e+00 1.082243e+00 1.092789e+00 1.103292e+00 1.113751e+00
## [96] 1.124167e+00 1.134538e+00 1.144864e+00 1.155145e+00 1.165381e+00
## [101] 1.175571e+00 1.185714e+00 1.195810e+00 1.205859e+00 1.215861e+00
## [106] 1.225814e+00 1.235719e+00 1.245576e+00 1.255383e+00 1.265140e+00
## [111] 1.274848e+00 1.284505e+00 1.294112e+00 1.303667e+00 1.313172e+00
## [116] 1.322624e+00 1.332024e+00 1.341371e+00 1.350666e+00 1.359907e+00
## [121] 1.369094e+00 1.378228e+00 1.387307e+00 1.396331e+00 1.405300e+00
## [126] 1.414214e+00 1.423071e+00 1.431873e+00 1.440618e+00 1.449306e+00
## [131] 1.457937e+00 1.466511e+00 1.475026e+00 1.483484e+00 1.491882e+00
## [136] 1.500222e+00 1.508503e+00 1.516724e+00 1.524885e+00 1.532986e+00
## [141] 1.541026e+00 1.549006e+00 1.556925e+00 1.564782e+00 1.572577e+00
## [146] 1.580310e+00 1.587981e+00 1.595589e+00 1.603134e+00 1.610616e+00
## [151] 1.618034e+00 1.625388e+00 1.632679e+00 1.639904e+00 1.647065e+00
## [156] 1.654161e+00 1.661192e+00 1.668157e+00 1.675056e+00 1.681889e+00
## [161] 1.688656e+00 1.695356e+00 1.701989e+00 1.708555e+00 1.715053e+00
## [166] 1.721484e+00 1.727847e+00 1.734141e+00 1.740368e+00 1.746525e+00
## [171] 1.752613e+00 1.758633e+00 1.764582e+00 1.770463e+00 1.776273e+00
## [176] 1.782013e+00 1.787683e+00 1.793282e+00 1.798811e+00 1.804268e+00
## [181] 1.809654e+00 1.814969e+00 1.820212e+00 1.825383e+00 1.830482e+00
## [186] 1.835509e+00 1.840464e+00 1.845345e+00 1.850154e+00 1.854890e+00
## [191] 1.859553e+00 1.864142e+00 1.868658e+00 1.873100e+00 1.877468e+00
## [196] 1.881762e+00 1.885981e+00 1.890126e+00 1.894197e+00 1.898192e+00
## [201] 1.902113e+00 1.905959e+00 1.909729e+00 1.913424e+00 1.917044e+00
## [206] 1.920587e+00 1.924055e+00 1.927447e+00 1.930763e+00 1.934003e+00
## [211] 1.937166e+00 1.940253e+00 1.943263e+00 1.946197e+00 1.949054e+00
## [216] 1.951834e+00 1.954536e+00 1.957162e+00 1.959710e+00 1.962181e+00
## [221] 1.964575e+00 1.966890e+00 1.969129e+00 1.971289e+00 1.973372e+00
## [226] 1.975377e+00 1.977303e+00 1.979152e+00 1.980923e+00 1.982615e+00
## [231] 1.984229e+00 1.985765e+00 1.987223e+00 1.988602e+00 1.989902e+00
## [236] 1.991124e+00 1.992267e+00 1.993332e+00 1.994318e+00 1.995225e+00
## [241] 1.996053e+00 1.996803e+00 1.997474e+00 1.998066e+00 1.998579e+00
## [246] 1.999013e+00 1.999368e+00 1.999645e+00 1.999842e+00 1.999961e+00
## [251] 2.000000e+00 1.999961e+00 1.999842e+00 1.999645e+00 1.999368e+00
## [256] 1.999013e+00 1.998579e+00 1.998066e+00 1.997474e+00 1.996803e+00
## [261] 1.996053e+00 1.995225e+00 1.994318e+00 1.993332e+00 1.992267e+00
## [266] 1.991124e+00 1.989902e+00 1.988602e+00 1.987223e+00 1.985765e+00
## [271] 1.984229e+00 1.982615e+00 1.980923e+00 1.979152e+00 1.977303e+00
## [276] 1.975377e+00 1.973372e+00 1.971289e+00 1.969129e+00 1.966890e+00
## [281] 1.964575e+00 1.962181e+00 1.959710e+00 1.957162e+00 1.954536e+00
## [286] 1.951834e+00 1.949054e+00 1.946197e+00 1.943263e+00 1.940253e+00
## [291] 1.937166e+00 1.934003e+00 1.930763e+00 1.927447e+00 1.924055e+00
## [296] 1.920587e+00 1.917044e+00 1.913424e+00 1.909729e+00 1.905959e+00
## [301] 1.902113e+00 1.898192e+00 1.894197e+00 1.890126e+00 1.885981e+00
## [306] 1.881762e+00 1.877468e+00 1.873100e+00 1.868658e+00 1.864142e+00
## [311] 1.859553e+00 1.854890e+00 1.850154e+00 1.845345e+00 1.840464e+00
## [316] 1.835509e+00 1.830482e+00 1.825383e+00 1.820212e+00 1.814969e+00
## [321] 1.809654e+00 1.804268e+00 1.798811e+00 1.793282e+00 1.787683e+00
## [326] 1.782013e+00 1.776273e+00 1.770463e+00 1.764582e+00 1.758633e+00
## [331] 1.752613e+00 1.746525e+00 1.740368e+00 1.734141e+00 1.727847e+00
## [336] 1.721484e+00 1.715053e+00 1.708555e+00 1.701989e+00 1.695356e+00
## [341] 1.688656e+00 1.681889e+00 1.675056e+00 1.668157e+00 1.661192e+00
## [346] 1.654161e+00 1.647065e+00 1.639904e+00 1.632679e+00 1.625388e+00
## [351] 1.618034e+00 1.610616e+00 1.603134e+00 1.595589e+00 1.587981e+00
## [356] 1.580310e+00 1.572577e+00 1.564782e+00 1.556925e+00 1.549006e+00
## [361] 1.541026e+00 1.532986e+00 1.524885e+00 1.516724e+00 1.508503e+00
## [366] 1.500222e+00 1.491882e+00 1.483484e+00 1.475026e+00 1.466511e+00
## [371] 1.457937e+00 1.449306e+00 1.440618e+00 1.431873e+00 1.423071e+00
## [376] 1.414214e+00 1.405300e+00 1.396331e+00 1.387307e+00 1.378228e+00
## [381] 1.369094e+00 1.359907e+00 1.350666e+00 1.341371e+00 1.332024e+00
## [386] 1.322624e+00 1.313172e+00 1.303667e+00 1.294112e+00 1.284505e+00
## [391] 1.274848e+00 1.265140e+00 1.255383e+00 1.245576e+00 1.235719e+00
## [396] 1.225814e+00 1.215861e+00 1.205859e+00 1.195810e+00 1.185714e+00
## [401] 1.175571e+00 1.165381e+00 1.155145e+00 1.144864e+00 1.134538e+00
## [406] 1.124167e+00 1.113751e+00 1.103292e+00 1.092789e+00 1.082243e+00
## [411] 1.071654e+00 1.061022e+00 1.050349e+00 1.039635e+00 1.028879e+00
## [416] 1.018083e+00 1.007246e+00 9.963702e-01 9.854547e-01 9.745003e-01
## [421] 9.635073e-01 9.524764e-01 9.414079e-01 9.303022e-01 9.191597e-01
## [426] 9.079810e-01 8.967664e-01 8.855165e-01 8.742315e-01 8.629121e-01
## [431] 8.515586e-01 8.401715e-01 8.287512e-01 8.172981e-01 8.058129e-01
## [436] 7.942958e-01 7.827473e-01 7.711680e-01 7.595582e-01 7.479184e-01
## [441] 7.362491e-01 7.245507e-01 7.128238e-01 7.010686e-01 6.892858e-01
## [446] 6.774758e-01 6.656391e-01 6.537761e-01 6.418872e-01 6.299730e-01
## [451] 6.180340e-01 6.060705e-01 5.940832e-01 5.820723e-01 5.700385e-01
## [456] 5.579822e-01 5.459039e-01 5.338040e-01 5.216830e-01 5.095415e-01
## [461] 4.973798e-01 4.851985e-01 4.729980e-01 4.607789e-01 4.485415e-01
## [466] 4.362865e-01 4.240142e-01 4.117252e-01 3.994200e-01 3.870989e-01
## [471] 3.747626e-01 3.624115e-01 3.500461e-01 3.376669e-01 3.252743e-01
## [476] 3.128689e-01 3.004512e-01 2.880216e-01 2.755806e-01 2.631287e-01
## [481] 2.506665e-01 2.381943e-01 2.257128e-01 2.132223e-01 2.007234e-01
## [486] 1.882166e-01 1.757024e-01 1.631812e-01 1.506536e-01 1.381201e-01
## [491] 1.255810e-01 1.130371e-01 1.004886e-01 8.793624e-02 7.538037e-02
## [496] 6.282152e-02 5.026019e-02 3.769688e-02 2.513208e-02 1.256629e-02
## [501] 2.449294e-16 -1.256629e-02 -2.513208e-02 -3.769688e-02 -5.026019e-02
## [506] -6.282152e-02 -7.538037e-02 -8.793624e-02 -1.004886e-01 -1.130371e-01
## [511] -1.255810e-01 -1.381201e-01 -1.506536e-01 -1.631812e-01 -1.757024e-01
## [516] -1.882166e-01 -2.007234e-01 -2.132223e-01 -2.257128e-01 -2.381943e-01
## [521] -2.506665e-01 -2.631287e-01 -2.755806e-01 -2.880216e-01 -3.004512e-01
## [526] -3.128689e-01 -3.252743e-01 -3.376669e-01 -3.500461e-01 -3.624115e-01
## [531] -3.747626e-01 -3.870989e-01 -3.994200e-01 -4.117252e-01 -4.240142e-01
## [536] -4.362865e-01 -4.485415e-01 -4.607789e-01 -4.729980e-01 -4.851985e-01
## [541] -4.973798e-01 -5.095415e-01 -5.216830e-01 -5.338040e-01 -5.459039e-01
## [546] -5.579822e-01 -5.700385e-01 -5.820723e-01 -5.940832e-01 -6.060705e-01
## [551] -6.180340e-01 -6.299730e-01 -6.418872e-01 -6.537761e-01 -6.656391e-01
## [556] -6.774758e-01 -6.892858e-01 -7.010686e-01 -7.128238e-01 -7.245507e-01
## [561] -7.362491e-01 -7.479184e-01 -7.595582e-01 -7.711680e-01 -7.827473e-01
## [566] -7.942958e-01 -8.058129e-01 -8.172981e-01 -8.287512e-01 -8.401715e-01
## [571] -8.515586e-01 -8.629121e-01 -8.742315e-01 -8.855165e-01 -8.967664e-01
## [576] -9.079810e-01 -9.191597e-01 -9.303022e-01 -9.414079e-01 -9.524764e-01
## [581] -9.635073e-01 -9.745003e-01 -9.854547e-01 -9.963702e-01 -1.007246e+00
## [586] -1.018083e+00 -1.028879e+00 -1.039635e+00 -1.050349e+00 -1.061022e+00
## [591] -1.071654e+00 -1.082243e+00 -1.092789e+00 -1.103292e+00 -1.113751e+00
## [596] -1.124167e+00 -1.134538e+00 -1.144864e+00 -1.155145e+00 -1.165381e+00
## [601] -1.175571e+00 -1.185714e+00 -1.195810e+00 -1.205859e+00 -1.215861e+00
## [606] -1.225814e+00 -1.235719e+00 -1.245576e+00 -1.255383e+00 -1.265140e+00
## [611] -1.274848e+00 -1.284505e+00 -1.294112e+00 -1.303667e+00 -1.313172e+00
## [616] -1.322624e+00 -1.332024e+00 -1.341371e+00 -1.350666e+00 -1.359907e+00
## [621] -1.369094e+00 -1.378228e+00 -1.387307e+00 -1.396331e+00 -1.405300e+00
## [626] -1.414214e+00 -1.423071e+00 -1.431873e+00 -1.440618e+00 -1.449306e+00
## [631] -1.457937e+00 -1.466511e+00 -1.475026e+00 -1.483484e+00 -1.491882e+00
## [636] -1.500222e+00 -1.508503e+00 -1.516724e+00 -1.524885e+00 -1.532986e+00
## [641] -1.541026e+00 -1.549006e+00 -1.556925e+00 -1.564782e+00 -1.572577e+00
## [646] -1.580310e+00 -1.587981e+00 -1.595589e+00 -1.603134e+00 -1.610616e+00
## [651] -1.618034e+00 -1.625388e+00 -1.632679e+00 -1.639904e+00 -1.647065e+00
## [656] -1.654161e+00 -1.661192e+00 -1.668157e+00 -1.675056e+00 -1.681889e+00
## [661] -1.688656e+00 -1.695356e+00 -1.701989e+00 -1.708555e+00 -1.715053e+00
## [666] -1.721484e+00 -1.727847e+00 -1.734141e+00 -1.740368e+00 -1.746525e+00
## [671] -1.752613e+00 -1.758633e+00 -1.764582e+00 -1.770463e+00 -1.776273e+00
## [676] -1.782013e+00 -1.787683e+00 -1.793282e+00 -1.798811e+00 -1.804268e+00
## [681] -1.809654e+00 -1.814969e+00 -1.820212e+00 -1.825383e+00 -1.830482e+00
## [686] -1.835509e+00 -1.840464e+00 -1.845345e+00 -1.850154e+00 -1.854890e+00
## [691] -1.859553e+00 -1.864142e+00 -1.868658e+00 -1.873100e+00 -1.877468e+00
## [696] -1.881762e+00 -1.885981e+00 -1.890126e+00 -1.894197e+00 -1.898192e+00
## [701] -1.902113e+00 -1.905959e+00 -1.909729e+00 -1.913424e+00 -1.917044e+00
## [706] -1.920587e+00 -1.924055e+00 -1.927447e+00 -1.930763e+00 -1.934003e+00
## [711] -1.937166e+00 -1.940253e+00 -1.943263e+00 -1.946197e+00 -1.949054e+00
## [716] -1.951834e+00 -1.954536e+00 -1.957162e+00 -1.959710e+00 -1.962181e+00
## [721] -1.964575e+00 -1.966890e+00 -1.969129e+00 -1.971289e+00 -1.973372e+00
## [726] -1.975377e+00 -1.977303e+00 -1.979152e+00 -1.980923e+00 -1.982615e+00
## [731] -1.984229e+00 -1.985765e+00 -1.987223e+00 -1.988602e+00 -1.989902e+00
## [736] -1.991124e+00 -1.992267e+00 -1.993332e+00 -1.994318e+00 -1.995225e+00
## [741] -1.996053e+00 -1.996803e+00 -1.997474e+00 -1.998066e+00 -1.998579e+00
## [746] -1.999013e+00 -1.999368e+00 -1.999645e+00 -1.999842e+00 -1.999961e+00
## [751] -2.000000e+00 -1.999961e+00 -1.999842e+00 -1.999645e+00 -1.999368e+00
## [756] -1.999013e+00 -1.998579e+00 -1.998066e+00 -1.997474e+00 -1.996803e+00
## [761] -1.996053e+00 -1.995225e+00 -1.994318e+00 -1.993332e+00 -1.992267e+00
## [766] -1.991124e+00 -1.989902e+00 -1.988602e+00 -1.987223e+00 -1.985765e+00
## [771] -1.984229e+00 -1.982615e+00 -1.980923e+00 -1.979152e+00 -1.977303e+00
## [776] -1.975377e+00 -1.973372e+00 -1.971289e+00 -1.969129e+00 -1.966890e+00
## [781] -1.964575e+00 -1.962181e+00 -1.959710e+00 -1.957162e+00 -1.954536e+00
## [786] -1.951834e+00 -1.949054e+00 -1.946197e+00 -1.943263e+00 -1.940253e+00
## [791] -1.937166e+00 -1.934003e+00 -1.930763e+00 -1.927447e+00 -1.924055e+00
## [796] -1.920587e+00 -1.917044e+00 -1.913424e+00 -1.909729e+00 -1.905959e+00
## [801] -1.902113e+00 -1.898192e+00 -1.894197e+00 -1.890126e+00 -1.885981e+00
## [806] -1.881762e+00 -1.877468e+00 -1.873100e+00 -1.868658e+00 -1.864142e+00
## [811] -1.859553e+00 -1.854890e+00 -1.850154e+00 -1.845345e+00 -1.840464e+00
## [816] -1.835509e+00 -1.830482e+00 -1.825383e+00 -1.820212e+00 -1.814969e+00
## [821] -1.809654e+00 -1.804268e+00 -1.798811e+00 -1.793282e+00 -1.787683e+00
## [826] -1.782013e+00 -1.776273e+00 -1.770463e+00 -1.764582e+00 -1.758633e+00
## [831] -1.752613e+00 -1.746525e+00 -1.740368e+00 -1.734141e+00 -1.727847e+00
## [836] -1.721484e+00 -1.715053e+00 -1.708555e+00 -1.701989e+00 -1.695356e+00
## [841] -1.688656e+00 -1.681889e+00 -1.675056e+00 -1.668157e+00 -1.661192e+00
## [846] -1.654161e+00 -1.647065e+00 -1.639904e+00 -1.632679e+00 -1.625388e+00
## [851] -1.618034e+00 -1.610616e+00 -1.603134e+00 -1.595589e+00 -1.587981e+00
## [856] -1.580310e+00 -1.572577e+00 -1.564782e+00 -1.556925e+00 -1.549006e+00
## [861] -1.541026e+00 -1.532986e+00 -1.524885e+00 -1.516724e+00 -1.508503e+00
## [866] -1.500222e+00 -1.491882e+00 -1.483484e+00 -1.475026e+00 -1.466511e+00
## [871] -1.457937e+00 -1.449306e+00 -1.440618e+00 -1.431873e+00 -1.423071e+00
## [876] -1.414214e+00 -1.405300e+00 -1.396331e+00 -1.387307e+00 -1.378228e+00
## [881] -1.369094e+00 -1.359907e+00 -1.350666e+00 -1.341371e+00 -1.332024e+00
## [886] -1.322624e+00 -1.313172e+00 -1.303667e+00 -1.294112e+00 -1.284505e+00
## [891] -1.274848e+00 -1.265140e+00 -1.255383e+00 -1.245576e+00 -1.235719e+00
## [896] -1.225814e+00 -1.215861e+00 -1.205859e+00 -1.195810e+00 -1.185714e+00
## [901] -1.175571e+00 -1.165381e+00 -1.155145e+00 -1.144864e+00 -1.134538e+00
## [906] -1.124167e+00 -1.113751e+00 -1.103292e+00 -1.092789e+00 -1.082243e+00
## [911] -1.071654e+00 -1.061022e+00 -1.050349e+00 -1.039635e+00 -1.028879e+00
## [916] -1.018083e+00 -1.007246e+00 -9.963702e-01 -9.854547e-01 -9.745003e-01
## [921] -9.635073e-01 -9.524764e-01 -9.414079e-01 -9.303022e-01 -9.191597e-01
## [926] -9.079810e-01 -8.967664e-01 -8.855165e-01 -8.742315e-01 -8.629121e-01
## [931] -8.515586e-01 -8.401715e-01 -8.287512e-01 -8.172981e-01 -8.058129e-01
## [936] -7.942958e-01 -7.827473e-01 -7.711680e-01 -7.595582e-01 -7.479184e-01
## [941] -7.362491e-01 -7.245507e-01 -7.128238e-01 -7.010686e-01 -6.892858e-01
## [946] -6.774758e-01 -6.656391e-01 -6.537761e-01 -6.418872e-01 -6.299730e-01
## [951] -6.180340e-01 -6.060705e-01 -5.940832e-01 -5.820723e-01 -5.700385e-01
## [956] -5.579822e-01 -5.459039e-01 -5.338040e-01 -5.216830e-01 -5.095415e-01
## [961] -4.973798e-01 -4.851985e-01 -4.729980e-01 -4.607789e-01 -4.485415e-01
## [966] -4.362865e-01 -4.240142e-01 -4.117252e-01 -3.994200e-01 -3.870989e-01
## [971] -3.747626e-01 -3.624115e-01 -3.500461e-01 -3.376669e-01 -3.252743e-01
## [976] -3.128689e-01 -3.004512e-01 -2.880216e-01 -2.755806e-01 -2.631287e-01
## [981] -2.506665e-01 -2.381943e-01 -2.257128e-01 -2.132223e-01 -2.007234e-01
## [986] -1.882166e-01 -1.757024e-01 -1.631812e-01 -1.506536e-01 -1.381201e-01
## [991] -1.255810e-01 -1.130371e-01 -1.004886e-01 -8.793624e-02 -7.538037e-02
## [996] -6.282152e-02 -5.026019e-02 -3.769688e-02 -2.513208e-02 -1.256629e-02
## [1001] -4.898587e-16x[1]## [1] 4y[1]## [1] 0c(x[1:3], y[1:3])## [1] 4.00000000 3.99992104 3.99968418 0.00000000 0.01256629 0.02513208plot(x, y, cex = 0.1)
s=rep(0,n)
for (i in 1:n) {
s[i] = -y[i]*x[i+1] + x[i]*y[i+1]
}
A=0.5*sum(s)
A## [1] 25.13258#The area of an ellipse according to formula: pi ab
pi*a*b## [1] 25.13274Compute the area of an ellipse by a planimeter with u = (0, x)
n=1000
a=4
b=2
t=seq(0, 2*pi, length = n+1)
x=a*cos(t)
y=b*sin(t)
s=rep(0,n)
for (i in 1:n) {
s[i] = x[i]*(y[i+1]-y[i])
}
A = sum(s)
A## [1] 25.13258plot(x, y, type = 'l')
ChatGPT code
a <- 3 # semi-major axis
b <- 2 # semi-minor axis
# Define the vector field F(x,y)
F <- function(x,y) {
c(-y/2, x/2)
}
# Calculate the line integral using the parametric equation of the ellipse
a <- 4 # semi-major axis
b <- 2 # semi-minor axis
# Define the vector field F(x,y)
F <- function(x,y) {
c(-y/2, x/2)
}
# Define the parametric equation of the ellipse
x <- function(t) a * cos(t)
y <- function(t) b * sin(t)
# Define the derivative of the parametric equation
dx <- function(t) -a * sin(t)
dy <- function(t) b * cos(t)
# Define the line integral using the parametric equation of the ellipse
integral <- 0
n <- 1000 # number of points
t <- seq(0, 2*pi, length.out = n)
for (i in 1:(n-1)) {
x1 <- x(t[i])
y1 <- y(t[i])
x2 <- x(t[i+1])
y2 <- y(t[i+1])
dx <- x2 - x1
dy <- y2 - y1
integral <- integral + F(x2,y2) %*% c(dx,dy)
}
# Calculate the area using Green's theorem
area <- integral
print(paste("The area of the ellipse is", area))## [1] "The area of the ellipse is 25.1325755310066"Exams: Midterm 2 Problem 5
del = 0.01
x = y = seq(-pi+del, pi-del, len = 401)
f = function(x,y){sin(x)*sin(y)}
z = outer(x, y, f)
par(mar=c(4.0, 4.0, 0.5, 0.5))
contour(x,y,z, col = 'blue',
levels = seq(-1,0, len =21),
lty='dotted',
xlab = 'x', ylab = 'y',
cex.lab =1.4, cex.axis = 1.4)
contour(x,y,z,
levels = seq(0,1, len =21),
col = 'red', add = TRUE)
x## [1] -3.13159265 -3.11593469 -3.10027673 -3.08461876 -3.06896080 -3.05330284
## [7] -3.03764487 -3.02198691 -3.00632895 -2.99067098 -2.97501302 -2.95935506
## [13] -2.94369709 -2.92803913 -2.91238117 -2.89672320 -2.88106524 -2.86540728
## [19] -2.84974931 -2.83409135 -2.81843339 -2.80277542 -2.78711746 -2.77145950
## [25] -2.75580154 -2.74014357 -2.72448561 -2.70882765 -2.69316968 -2.67751172
## [31] -2.66185376 -2.64619579 -2.63053783 -2.61487987 -2.59922190 -2.58356394
## [37] -2.56790598 -2.55224801 -2.53659005 -2.52093209 -2.50527412 -2.48961616
## [43] -2.47395820 -2.45830023 -2.44264227 -2.42698431 -2.41132634 -2.39566838
## [49] -2.38001042 -2.36435245 -2.34869449 -2.33303653 -2.31737856 -2.30172060
## [55] -2.28606264 -2.27040467 -2.25474671 -2.23908875 -2.22343078 -2.20777282
## [61] -2.19211486 -2.17645689 -2.16079893 -2.14514097 -2.12948300 -2.11382504
## [67] -2.09816708 -2.08250911 -2.06685115 -2.05119319 -2.03553522 -2.01987726
## [73] -2.00421930 -1.98856134 -1.97290337 -1.95724541 -1.94158745 -1.92592948
## [79] -1.91027152 -1.89461356 -1.87895559 -1.86329763 -1.84763967 -1.83198170
## [85] -1.81632374 -1.80066578 -1.78500781 -1.76934985 -1.75369189 -1.73803392
## [91] -1.72237596 -1.70671800 -1.69106003 -1.67540207 -1.65974411 -1.64408614
## [97] -1.62842818 -1.61277022 -1.59711225 -1.58145429 -1.56579633 -1.55013836
## [103] -1.53448040 -1.51882244 -1.50316447 -1.48750651 -1.47184855 -1.45619058
## [109] -1.44053262 -1.42487466 -1.40921669 -1.39355873 -1.37790077 -1.36224280
## [115] -1.34658484 -1.33092688 -1.31526891 -1.29961095 -1.28395299 -1.26829502
## [121] -1.25263706 -1.23697910 -1.22132113 -1.20566317 -1.19000521 -1.17434725
## [127] -1.15868928 -1.14303132 -1.12737336 -1.11171539 -1.09605743 -1.08039947
## [133] -1.06474150 -1.04908354 -1.03342558 -1.01776761 -1.00210965 -0.98645169
## [139] -0.97079372 -0.95513576 -0.93947780 -0.92381983 -0.90816187 -0.89250391
## [145] -0.87684594 -0.86118798 -0.84553002 -0.82987205 -0.81421409 -0.79855613
## [151] -0.78289816 -0.76724020 -0.75158224 -0.73592427 -0.72026631 -0.70460835
## [157] -0.68895038 -0.67329242 -0.65763446 -0.64197649 -0.62631853 -0.61066057
## [163] -0.59500260 -0.57934464 -0.56368668 -0.54802871 -0.53237075 -0.51671279
## [169] -0.50105482 -0.48539686 -0.46973890 -0.45408093 -0.43842297 -0.42276501
## [175] -0.40710704 -0.39144908 -0.37579112 -0.36013316 -0.34447519 -0.32881723
## [181] -0.31315927 -0.29750130 -0.28184334 -0.26618538 -0.25052741 -0.23486945
## [187] -0.21921149 -0.20355352 -0.18789556 -0.17223760 -0.15657963 -0.14092167
## [193] -0.12526371 -0.10960574 -0.09394778 -0.07828982 -0.06263185 -0.04697389
## [199] -0.03131593 -0.01565796 0.00000000 0.01565796 0.03131593 0.04697389
## [205] 0.06263185 0.07828982 0.09394778 0.10960574 0.12526371 0.14092167
## [211] 0.15657963 0.17223760 0.18789556 0.20355352 0.21921149 0.23486945
## [217] 0.25052741 0.26618538 0.28184334 0.29750130 0.31315927 0.32881723
## [223] 0.34447519 0.36013316 0.37579112 0.39144908 0.40710704 0.42276501
## [229] 0.43842297 0.45408093 0.46973890 0.48539686 0.50105482 0.51671279
## [235] 0.53237075 0.54802871 0.56368668 0.57934464 0.59500260 0.61066057
## [241] 0.62631853 0.64197649 0.65763446 0.67329242 0.68895038 0.70460835
## [247] 0.72026631 0.73592427 0.75158224 0.76724020 0.78289816 0.79855613
## [253] 0.81421409 0.82987205 0.84553002 0.86118798 0.87684594 0.89250391
## [259] 0.90816187 0.92381983 0.93947780 0.95513576 0.97079372 0.98645169
## [265] 1.00210965 1.01776761 1.03342558 1.04908354 1.06474150 1.08039947
## [271] 1.09605743 1.11171539 1.12737336 1.14303132 1.15868928 1.17434725
## [277] 1.19000521 1.20566317 1.22132113 1.23697910 1.25263706 1.26829502
## [283] 1.28395299 1.29961095 1.31526891 1.33092688 1.34658484 1.36224280
## [289] 1.37790077 1.39355873 1.40921669 1.42487466 1.44053262 1.45619058
## [295] 1.47184855 1.48750651 1.50316447 1.51882244 1.53448040 1.55013836
## [301] 1.56579633 1.58145429 1.59711225 1.61277022 1.62842818 1.64408614
## [307] 1.65974411 1.67540207 1.69106003 1.70671800 1.72237596 1.73803392
## [313] 1.75369189 1.76934985 1.78500781 1.80066578 1.81632374 1.83198170
## [319] 1.84763967 1.86329763 1.87895559 1.89461356 1.91027152 1.92592948
## [325] 1.94158745 1.95724541 1.97290337 1.98856134 2.00421930 2.01987726
## [331] 2.03553522 2.05119319 2.06685115 2.08250911 2.09816708 2.11382504
## [337] 2.12948300 2.14514097 2.16079893 2.17645689 2.19211486 2.20777282
## [343] 2.22343078 2.23908875 2.25474671 2.27040467 2.28606264 2.30172060
## [349] 2.31737856 2.33303653 2.34869449 2.36435245 2.38001042 2.39566838
## [355] 2.41132634 2.42698431 2.44264227 2.45830023 2.47395820 2.48961616
## [361] 2.50527412 2.52093209 2.53659005 2.55224801 2.56790598 2.58356394
## [367] 2.59922190 2.61487987 2.63053783 2.64619579 2.66185376 2.67751172
## [373] 2.69316968 2.70882765 2.72448561 2.74014357 2.75580154 2.77145950
## [379] 2.78711746 2.80277542 2.81843339 2.83409135 2.84974931 2.86540728
## [385] 2.88106524 2.89672320 2.91238117 2.92803913 2.94369709 2.95935506
## [391] 2.97501302 2.99067098 3.00632895 3.02198691 3.03764487 3.05330284
## [397] 3.06896080 3.08461876 3.10027673 3.11593469 3.13159265Exercise 10.6(a): Model flower pedals
par(mar=c(4,4,2,0.2))
# eight petals
a = 3
k = 4
n = 1000
ome = 2
t = seq(0, 2*pi, len=n)
theta = ome*t
r = a*cos(k*theta)
x = r*cos(theta)
y = r*sin(theta)
plot(x, y, type = 'l', lwd = 4, col = 'pink',
main = 'An eight-petal rose')
x## [1] 3.000000000 2.995966240 2.983878210 2.963775607 2.935724434
## [6] 2.899816753 2.856170354 2.804928317 2.746258495 2.680352894
## [11] 2.607426974 2.527718860 2.441488473 2.349016578 2.250603765
## [16] 2.146569347 2.037250199 1.922999536 1.804185620 1.681190433
## [21] 1.554408282 1.424244378 1.291113366 1.155437832 1.017646776
## [26] 0.878174068 0.737456897 0.595934196 0.454045080 0.312227279
## [31] 0.170915583 0.030540302 -0.108474253 -0.245711263 -0.380762855
## [36] -0.513231529 -0.642731533 -0.768890190 -0.891349176 -1.009765736
## [41] -1.123813835 -1.233185252 -1.337590587 -1.436760206 -1.530445105
## [46] -1.618417688 -1.700472462 -1.776426652 -1.846120718 -1.909418787
## [51] -1.966208989 -2.016403706 -2.059939721 -2.096778277 -2.126905038
## [56] -2.150329959 -2.167087067 -2.177234140 -2.180852306 -2.178045547
## [61] -2.168940123 -2.153683904 -2.132445629 -2.105414086 -2.072797211
## [66] -2.034821126 -1.991729105 -1.943780480 -1.891249489 -1.834424070
## [71] -1.773604609 -1.709102644 -1.641239532 -1.570345084 -1.496756175
## [76] -1.420815325 -1.342869281 -1.263267572 -1.182361068 -1.100500548
## [81] -1.018035262 -0.935311516 -0.852671274 -0.770450785 -0.688979238
## [86] -0.608577458 -0.529556634 -0.452217103 -0.376847177 -0.303722029
## [91] -0.233102635 -0.165234788 -0.100348165 -0.038655482 0.019648291
## [96] 0.074386625 0.125402060 0.172556737 0.215732858 0.254833050
## [101] 0.289780648 0.320519884 0.347015996 0.369255235 0.387244795
## [106] 0.401012645 0.410607281 0.416097386 0.417571405 0.415137043
## [111] 0.408920671 0.399066665 0.385736663 0.369108751 0.349376580
## [116] 0.326748420 0.301446146 0.273704178 0.243768361 0.211894796
## [121] 0.178348633 0.143402827 0.107336855 0.070435412 0.032987080
## [126] -0.004717011 -0.042384544 -0.079723370 -0.116442871 -0.152255311
## [131] -0.186877174 -0.220030493 -0.251444147 -0.280855131 -0.308009797
## [136] -0.332665048 -0.354589498 -0.373564575 -0.389385572 -0.401862642
## [141] -0.410821732 -0.416105446 -0.417573842 -0.415105156 -0.408596446
## [146] -0.397964163 -0.383144636 -0.364094471 -0.340790872 -0.313231870
## [151] -0.281436462 -0.245444669 -0.205317494 -0.161136802 -0.113005101
## [156] -0.061045244 -0.005400033 0.053768248 0.116278401 0.181930948
## [161] 0.250508859 0.321778343 0.395489725 0.471378381 0.549165754
## [166] 0.628560418 0.709259212 0.790948420 0.873305007 0.955997895
## [171] 1.038689278 1.121035978 1.202690820 1.283304039 1.362524699
## [176] 1.440002128 1.515387359 1.588334569 1.658502512 1.725555951
## [181] 1.789167057 1.849016801 1.904796310 1.956208189 2.002967811
## [186] 2.044804552 2.081462990 2.112704038 2.138306026 2.158065717
## [191] 2.171799254 2.179343043 2.180554548 2.175313021 2.163520140
## [196] 2.145100574 2.120002447 2.088197725 2.049682508 2.004477228
## [201] 1.952626758 1.894200425 1.829291927 1.758019161 1.680523952
## [206] 1.596971694 1.507550893 1.412472627 1.311969915 1.206296994
## [211] 1.095728524 0.980558699 0.861100296 0.737683630 0.610655462
## [216] 0.480377819 0.347226768 0.211591127 0.073871119 -0.065523013
## [221] -0.206172437 -0.347651201 -0.489527733 -0.631366364 -0.772728869
## [226] -0.913176033 -1.052269211 -1.189571899 -1.324651297 -1.457079864
## [231] -1.586436860 -1.712309858 -1.834296236 -1.952004637 -2.065056381
## [236] -2.173086843 -2.275746776 -2.372703583 -2.463642525 -2.548267869
## [241] -2.626303966 -2.697496260 -2.761612212 -2.818442156 -2.867800062
## [246] -2.909524215 -2.943477814 -2.969549462 -2.987653586 -2.997730738
## [251] -2.999747825 -2.993698221 -2.979601795 -2.957504839 -2.927479897
## [256] -2.889625498 -2.844065798 -2.790950124 -2.730452428 -2.662770648
## [261] -2.588125989 -2.506762108 -2.418944229 -2.324958174 -2.225109317
## [266] -2.119721475 -2.009135725 -1.893709165 -1.773813615 -1.649834269
## [271] -1.522168298 -1.391223408 -1.257416372 -1.121171522 -0.982919221
## [276] -0.843094316 -0.702134578 -0.560479134 -0.418566898 -0.276835011
## [281] -0.135717285 0.004357326 0.142966247 0.279694759 0.414137452
## [286] 0.545899634 0.674598701 0.799865445 0.921345324 1.038699656
## [291] 1.151606762 1.259763033 1.362883925 1.460704885 1.552982191
## [296] 1.639493708 1.720039572 1.794442771 1.862549643 1.924230288
## [301] 1.979378882 2.027913895 2.069778224 2.104939223 2.133388646
## [306] 2.155142490 2.170240753 2.178747091 2.180748396 2.176354278
## [311] 2.165696464 2.148928114 2.126223060 2.097774963 2.063796398
## [316] 2.024517875 1.980186789 1.931066306 1.877434206 1.819581657
## [321] 1.757811957 1.692439224 1.623787058 1.552187169 1.477977973
## [326] 1.401503184 1.323110379 1.243149555 1.161971698 1.079927335
## [331] 0.997365114 0.914630384 0.832063810 0.750000000 0.668766173
## [336] 0.588680861 0.510052653 0.433178989 0.358344997 0.285822398
## [341] 0.215868462 0.148725037 0.084617637 0.023754612 -0.033673615
## [346] -0.087495229 -0.137557622 -0.183727912 -0.225893375 -0.263961792
## [351] -0.297861708 -0.327542601 -0.352974964 -0.374150294 -0.391080997
## [356] -0.403800202 -0.412361486 -0.416838516 -0.417324603 -0.413932176
## [361] -0.406792172 -0.396053357 -0.381881558 -0.364458839 -0.343982597
## [366] -0.320664598 -0.294729958 -0.266416060 -0.235971421 -0.203654522
## [371] -0.169732583 -0.134480308 -0.098178603 -0.061113261 -0.023573626
## [376] 0.014148747 0.051761455 0.088972606 0.125492175 0.161033356
## [381] 0.195313897 0.228057423 0.258994721 0.287865013 0.314417177
## [386] 0.338410938 0.359618012 0.377823197 0.392825408 0.404438665
## [391] 0.412492998 0.416835304 0.417330120 0.413860329 0.406327790
## [396] 0.394653880 0.378779961 0.358667765 0.334299682 0.305678977
## [401] 0.272829900 0.235797725 0.194648685 0.149469830 0.100368786
## [406] 0.047473433 -0.009068509 -0.069089971 -0.132405122 -0.198810025
## [411] -0.268083384 -0.339987350 -0.414268420 -0.490658392 -0.568875387
## [416] -0.648624938 -0.729601131 -0.811487805 -0.893959792 -0.976684206
## [421] -1.059321772 -1.141528180 -1.222955477 -1.303253470 -1.382071156
## [426] -1.459058153 -1.533866145 -1.606150319 -1.675570800 -1.741794069
## [431] -1.804494370 -1.863355086 -1.918070091 -1.968345064 -2.013898762
## [436] -2.054464255 -2.089790097 -2.119641458 -2.143801184 -2.162070796
## [441] -2.174271424 -2.180244665 -2.179853367 -2.172982329 -2.159538931
## [446] -2.139453663 -2.112680577 -2.079197649 -2.039007047 -1.992135307
## [451] -1.938633418 -1.878576810 -1.812065252 -1.739222652 -1.660196764
## [456] -1.575158805 -1.484302981 -1.387845917 -1.286026005 -1.179102669
## [461] -1.067355537 -0.951083541 -0.830603943 -0.706251276 -0.578376229
## [466] -0.447344458 -0.313535339 -0.177340670 -0.039163310 0.100584213
## [471] 0.241481151 0.383100004 0.525008031 0.666768779 0.807943630
## [476] 0.948093364 1.086779723 1.223566985 1.358023517 1.489723337
## [481] 1.618247641 1.743186314 1.864139416 1.980718628 2.092548656
## [486] 2.199268599 2.300533255 2.396014379 2.485401880 2.568404946
## [491] 2.644753113 2.714197242 2.776510437 2.831488873 2.878952538
## [496] 2.918745894 2.950738445 2.974825219 2.990927147 2.998991353
## [501] 2.998991353 2.990927147 2.974825219 2.950738445 2.918745894
## [506] 2.878952538 2.831488873 2.776510437 2.714197242 2.644753113
## [511] 2.568404946 2.485401880 2.396014379 2.300533255 2.199268599
## [516] 2.092548656 1.980718628 1.864139416 1.743186314 1.618247641
## [521] 1.489723337 1.358023517 1.223566985 1.086779723 0.948093364
## [526] 0.807943630 0.666768779 0.525008031 0.383100004 0.241481151
## [531] 0.100584213 -0.039163310 -0.177340670 -0.313535339 -0.447344458
## [536] -0.578376229 -0.706251276 -0.830603943 -0.951083541 -1.067355537
## [541] -1.179102669 -1.286026005 -1.387845917 -1.484302981 -1.575158805
## [546] -1.660196764 -1.739222652 -1.812065252 -1.878576810 -1.938633418
## [551] -1.992135307 -2.039007047 -2.079197649 -2.112680577 -2.139453663
## [556] -2.159538931 -2.172982329 -2.179853367 -2.180244665 -2.174271424
## [561] -2.162070796 -2.143801184 -2.119641458 -2.089790097 -2.054464255
## [566] -2.013898762 -1.968345064 -1.918070091 -1.863355086 -1.804494370
## [571] -1.741794069 -1.675570800 -1.606150319 -1.533866145 -1.459058153
## [576] -1.382071156 -1.303253470 -1.222955477 -1.141528180 -1.059321772
## [581] -0.976684206 -0.893959792 -0.811487805 -0.729601131 -0.648624938
## [586] -0.568875387 -0.490658392 -0.414268420 -0.339987350 -0.268083384
## [591] -0.198810025 -0.132405122 -0.069089971 -0.009068509 0.047473433
## [596] 0.100368786 0.149469830 0.194648685 0.235797725 0.272829900
## [601] 0.305678977 0.334299682 0.358667765 0.378779961 0.394653880
## [606] 0.406327790 0.413860329 0.417330120 0.416835304 0.412492998
## [611] 0.404438665 0.392825408 0.377823197 0.359618012 0.338410938
## [616] 0.314417177 0.287865013 0.258994721 0.228057423 0.195313897
## [621] 0.161033356 0.125492175 0.088972606 0.051761455 0.014148747
## [626] -0.023573626 -0.061113261 -0.098178603 -0.134480308 -0.169732583
## [631] -0.203654522 -0.235971421 -0.266416060 -0.294729958 -0.320664598
## [636] -0.343982597 -0.364458839 -0.381881558 -0.396053357 -0.406792172
## [641] -0.413932176 -0.417324603 -0.416838516 -0.412361486 -0.403800202
## [646] -0.391080997 -0.374150294 -0.352974964 -0.327542601 -0.297861708
## [651] -0.263961792 -0.225893375 -0.183727912 -0.137557622 -0.087495229
## [656] -0.033673615 0.023754612 0.084617637 0.148725037 0.215868462
## [661] 0.285822398 0.358344997 0.433178989 0.510052653 0.588680861
## [666] 0.668766173 0.750000000 0.832063810 0.914630384 0.997365114
## [671] 1.079927335 1.161971698 1.243149555 1.323110379 1.401503184
## [676] 1.477977973 1.552187169 1.623787058 1.692439224 1.757811957
## [681] 1.819581657 1.877434206 1.931066306 1.980186789 2.024517875
## [686] 2.063796398 2.097774963 2.126223060 2.148928114 2.165696464
## [691] 2.176354278 2.180748396 2.178747091 2.170240753 2.155142490
## [696] 2.133388646 2.104939223 2.069778224 2.027913895 1.979378882
## [701] 1.924230288 1.862549643 1.794442771 1.720039572 1.639493708
## [706] 1.552982191 1.460704885 1.362883925 1.259763033 1.151606762
## [711] 1.038699656 0.921345324 0.799865445 0.674598701 0.545899634
## [716] 0.414137452 0.279694759 0.142966247 0.004357326 -0.135717285
## [721] -0.276835011 -0.418566898 -0.560479134 -0.702134578 -0.843094316
## [726] -0.982919221 -1.121171522 -1.257416372 -1.391223408 -1.522168298
## [731] -1.649834269 -1.773813615 -1.893709165 -2.009135725 -2.119721475
## [736] -2.225109317 -2.324958174 -2.418944229 -2.506762108 -2.588125989
## [741] -2.662770648 -2.730452428 -2.790950124 -2.844065798 -2.889625498
## [746] -2.927479897 -2.957504839 -2.979601795 -2.993698221 -2.999747825
## [751] -2.997730738 -2.987653586 -2.969549462 -2.943477814 -2.909524215
## [756] -2.867800062 -2.818442156 -2.761612212 -2.697496260 -2.626303966
## [761] -2.548267869 -2.463642525 -2.372703583 -2.275746776 -2.173086843
## [766] -2.065056381 -1.952004637 -1.834296236 -1.712309858 -1.586436860
## [771] -1.457079864 -1.324651297 -1.189571899 -1.052269211 -0.913176033
## [776] -0.772728869 -0.631366364 -0.489527733 -0.347651201 -0.206172437
## [781] -0.065523013 0.073871119 0.211591127 0.347226768 0.480377819
## [786] 0.610655462 0.737683630 0.861100296 0.980558699 1.095728524
## [791] 1.206296994 1.311969915 1.412472627 1.507550893 1.596971694
## [796] 1.680523952 1.758019161 1.829291927 1.894200425 1.952626758
## [801] 2.004477228 2.049682508 2.088197725 2.120002447 2.145100574
## [806] 2.163520140 2.175313021 2.180554548 2.179343043 2.171799254
## [811] 2.158065717 2.138306026 2.112704038 2.081462990 2.044804552
## [816] 2.002967811 1.956208189 1.904796310 1.849016801 1.789167057
## [821] 1.725555951 1.658502512 1.588334569 1.515387359 1.440002128
## [826] 1.362524699 1.283304039 1.202690820 1.121035978 1.038689278
## [831] 0.955997895 0.873305007 0.790948420 0.709259212 0.628560418
## [836] 0.549165754 0.471378381 0.395489725 0.321778343 0.250508859
## [841] 0.181930948 0.116278401 0.053768248 -0.005400033 -0.061045244
## [846] -0.113005101 -0.161136802 -0.205317494 -0.245444669 -0.281436462
## [851] -0.313231870 -0.340790872 -0.364094471 -0.383144636 -0.397964163
## [856] -0.408596446 -0.415105156 -0.417573842 -0.416105446 -0.410821732
## [861] -0.401862642 -0.389385572 -0.373564575 -0.354589498 -0.332665048
## [866] -0.308009797 -0.280855131 -0.251444147 -0.220030493 -0.186877174
## [871] -0.152255311 -0.116442871 -0.079723370 -0.042384544 -0.004717011
## [876] 0.032987080 0.070435412 0.107336855 0.143402827 0.178348633
## [881] 0.211894796 0.243768361 0.273704178 0.301446146 0.326748420
## [886] 0.349376580 0.369108751 0.385736663 0.399066665 0.408920671
## [891] 0.415137043 0.417571405 0.416097386 0.410607281 0.401012645
## [896] 0.387244795 0.369255235 0.347015996 0.320519884 0.289780648
## [901] 0.254833050 0.215732858 0.172556737 0.125402060 0.074386625
## [906] 0.019648291 -0.038655482 -0.100348165 -0.165234788 -0.233102635
## [911] -0.303722029 -0.376847177 -0.452217103 -0.529556634 -0.608577458
## [916] -0.688979238 -0.770450785 -0.852671274 -0.935311516 -1.018035262
## [921] -1.100500548 -1.182361068 -1.263267572 -1.342869281 -1.420815325
## [926] -1.496756175 -1.570345084 -1.641239532 -1.709102644 -1.773604609
## [931] -1.834424070 -1.891249489 -1.943780480 -1.991729105 -2.034821126
## [936] -2.072797211 -2.105414086 -2.132445629 -2.153683904 -2.168940123
## [941] -2.178045547 -2.180852306 -2.177234140 -2.167087067 -2.150329959
## [946] -2.126905038 -2.096778277 -2.059939721 -2.016403706 -1.966208989
## [951] -1.909418787 -1.846120718 -1.776426652 -1.700472462 -1.618417688
## [956] -1.530445105 -1.436760206 -1.337590587 -1.233185252 -1.123813835
## [961] -1.009765736 -0.891349176 -0.768890190 -0.642731533 -0.513231529
## [966] -0.380762855 -0.245711263 -0.108474253 0.030540302 0.170915583
## [971] 0.312227279 0.454045080 0.595934196 0.737456897 0.878174068
## [976] 1.017646776 1.155437832 1.291113366 1.424244378 1.554408282
## [981] 1.681190433 1.804185620 1.922999536 2.037250199 2.146569347
## [986] 2.250603765 2.349016578 2.441488473 2.527718860 2.607426974
## [991] 2.680352894 2.746258495 2.804928317 2.856170354 2.899816753
## [996] 2.935724434 2.963775607 2.983878210 2.995966240 3.000000000plot(x[1:125], y[1:125], type = 'l', lwd = 4, col = 'pink',
main = 'An eight-petal rose')
#One petal
plot(x[33:94], y[33:94],
xlim = c(-3, 3), ylim = c(-3, 3),
type = 'l', lwd = 1, col = 'red',
main = 'An eight-petal rose')
#Apply Green's theorem
xpedal = x[33:94]
ypedal = y[33:94]
m = length(xpedal)
s=rep(0,m)
for (i in 1:(m-1)){s[i] = xpedal[i]*(ypedal[i+1] - ypedal[i])}
A = sum(s)
A## [1] 1.6099648*A #Output the result## [1] 12.87971###Jingle Bells Music in R by Keith McNulty in 2018 ####https://paulvanderlaken.com/2017/12/18/jingle-bells-in-r/
if(!"dplyr" %in% installed.packages()) install.packages("dplyr")
if(!"audio" %in% installed.packages()) install.packages("audio")
library("dplyr")
library("audio")
notes <- c(A = 0, B = 2, C = 3, D = 5, E = 7, F = 8, G = 10)
pitch <- paste("E E E",
"E E E",
"E G C D",
"E",
"F F F F",
"F E E E",
"E D D E",
"D G",
"E E E",
"E E E",
"E G C D",
"E",
"F F F F",
"F E E E E",
"G G F D",
"C",
"G3 E D C",
"G3",
"G3 G3 G3 E D C",
"A3",
"A3 F E D",
"B3",
"G G F D",
"E",
"G3 E D C",
"G3",
"G3 E D C",
"A3 A3",
"A3 F E D",
"G G G G A G F D",
"C C5 B A G F G",
"E E E G C D",
"E E E G C D",
"E F G A C E D F",
"E C D E F G A G",
"F F F F F F",
"F E E E E E",
"E D D D D E",
"D D E F G F E D",
"E E E G C D",
"E E E G C D",
"E F G A C E D F",
"E C D E F G A G",
"F F F F F F",
"F E E E E E",
"G C5 B A G F E D",
"C C E G C5")
duration <- c(1, 1, 2,
1, 1, 2,
1, 1, 1.5, 0.5,
4,
1, 1, 1, 1,
1, 1, 1, 1,
1, 1, 1, 1,
2, 2,
1, 1, 2,
1, 1, 2,
1, 1, 1.5, 0.5,
4,
1, 1, 1, 1,
1, 1, 1, 0.5, 0.5,
1, 1, 1, 1,
4,
1, 1, 1, 1,
3, .5, .5,
1, 1, 1, 1,
4,
1, 1, 1, 1,
4,
1, 1, 1, 1,
4,
1, 1, 1, 1,
4,
1, 1, 1, 1,
3, 1,
1, 1, 1, 1,
1, 1, 1, 1,
1, 1, 1, 1,
1, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
1, 1, 0.5, 0.5, 0.5, 0.5,
1, 1, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
1, 0.5, 0.5, 1, 0.5, 0.5,
1, 0.5, 0.5, 1, 0.5, 0.5,
1, 0.5, 0.5, 0.5, 0.5, 1,
1, 0.33, 0.33, 0.33, 1, 0.33, 0.33, 0.33,
1, 1, 0.5, 0.5, 0.5, 0.5,
1, 1, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
1, 0.5, 0.5, 1, 0.5, 0.5,
1, 0.5, 0.5, 1, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
1, 0.33, 0.33, 0.33, 2)
jbells <- data_frame(pitch = strsplit(pitch, " ")[[1]],
duration = duration)
jbells <- jbells %>%
mutate(octave = substring(pitch, nchar(pitch)) %>%
{suppressWarnings(as.numeric(.))} %>%
ifelse(is.na(.), 4, .),
note = notes[substr(pitch, 1, 1)],
note = note + grepl("#", pitch) -
grepl("b", pitch) + octave * 12 +
12 * (note < 3),
freq = 2 ^ ((note - 60) / 12) * 440)
tempo <- 250
sample_rate <- 44100
make_sine <- function(freq, duration) {
wave <- sin(seq(0, duration / tempo * 60, 1 / sample_rate) *
freq * 2 * pi)
fade <- seq(0, 1, 50 / sample_rate)
wave * c(fade, rep(1, length(wave) - 2 * length(fade)), rev(fade))
}
jbells_wave <- mapply(make_sine, jbells$freq, jbells$duration) %>%
do.call("c", .)
play(jbells_wave)Ivanna’s Jingle Bell, Jingle Bells Music in R by Keith McNulty in 2018
#####https://paulvanderlaken.com/2017/12/18/jingle-bells-in-r/
if(!"dplyr" %in% installed.packages()) install.packages("dplyr")
if(!"audio" %in% installed.packages()) install.packages("audio")
library("dplyr")
library("audio")
notes <- c(A = 0, B = 2, C = 3, D = 5, E = 7, F = 8, G = 10)
pitch <- paste("C E C",
"E E E",
"E G C D",
"E",
"F F F F",
"F E E E",
"E D D E",
"D G",
"E E E",
"E E E",
"E G C D",
"E",
"F F F F",
"F E E E E",
"G G F D",
"C",
"G3 E D C",
"G3",
"G3 G3 G3 E D C",
"A3",
"A3 F E D",
"B3",
"G G F D",
"E",
"G3 E D C",
"G3",
"G3 E D C",
"A3 A3",
"A3 F E D",
"G G G G A G F D",
"C C5 B A G F G",
"E E E G C D",
"E E E G C D",
"E F G A C E D F",
"E C D E F G A G",
"F F F F F F",
"F E E E E E",
"E D D D D E",
"D D E F G F E D",
"E E E G C D",
"E E E G C D",
"E F G A C E D F",
"E C D E F G A G",
"F F F F F F",
"F E E E E E",
"G C5 B A G F E D",
"C C E G C5")
duration <- c(1, 3, 2,
1, 3, 2,
1, 4, 1.5, 2.5,
4,
1, 1, 1, 1,
1, 1, 1, 1,
1, 1, 1, 1,
2, 2,
1, 1, 2,
1, 1, 2,
1, 1, 1.5, 0.5,
4,
1, 1, 1, 1,
1, 1, 1, 0.5, 0.5,
1, 1, 1, 1,
4,
1, 1, 1, 1,
3, .5, .5,
1, 1, 1, 1,
4,
1, 1, 1, 1,
4,
1, 1, 1, 1,
4,
1, 1, 1, 1,
4,
1, 1, 1, 1,
3, 1,
1, 1, 1, 1,
1, 1, 1, 1,
1, 1, 1, 1,
1, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
1, 1, 0.5, 0.5, 0.5, 0.5,
1, 1, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
1, 0.5, 0.5, 1, 0.5, 0.5,
1, 0.5, 0.5, 1, 0.5, 0.5,
1, 0.5, 0.5, 0.5, 0.5, 1,
1, 0.33, 0.33, 0.33, 1, 0.33, 0.33, 0.33,
1, 1, 0.5, 0.5, 0.5, 0.5,
1, 1, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
1, 0.5, 0.5, 1, 0.5, 0.5,
1, 0.5, 0.5, 1, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
1, 0.33, 0.33, 0.33, 2)
jbells <- data_frame(pitch = strsplit(pitch, " ")[[1]],
duration = duration)
jbells <- jbells %>%
mutate(octave = substring(pitch, nchar(pitch)) %>%
{suppressWarnings(as.numeric(.))} %>%
ifelse(is.na(.), 4, .),
note = notes[substr(pitch, 1, 1)],
note = note + grepl("#", pitch) -
grepl("b", pitch) + octave * 12 +
12 * (note < 3),
freq = 2 ^ ((note - 60) / 12) * 440)
tempo <- 250
sample_rate <- 44100
make_sine <- function(freq, duration) {
wave <- sin(seq(0, duration / tempo * 60, 1 / sample_rate) *
freq * 2 * pi)
fade <- seq(0, 1, 50 / sample_rate)
wave * c(fade, rep(1, length(wave) - 2 * length(fade)), rev(fade))
}
jbells_wave <- mapply(make_sine, jbells$freq, jbells$duration) %>%
do.call("c", .)
play(jbells_wave)Revised Jingle Bell, Jingle Bells Music in R by Keith McNulty in 2018
####https://paulvanderlaken.com/2017/12/18/jingle-bells-in-r/
if(!"dplyr" %in% installed.packages()) install.packages("dplyr")
if(!"audio" %in% installed.packages()) install.packages("audio")
library("dplyr")
library("audio")
notes <- c(A = 0, B = 2, C = 3, D = 5, E = 7, F = 8, G = 10)
pitch <- paste("F C F",
"G C G",
"G G C D",
"E",
"F F F F",
"G B B B",
"E D D E",
"D G",
"E E E",
"E E E",
"E G C D",
"E",
"F F F F",
"F E E E E",
"G G F D",
"C",
"G3 E D C",
"G3",
"G3 G3 G3 E D C",
"A3",
"A3 F E D",
"B3",
"G G F D",
"E",
"G3 E D C",
"G3",
"G3 E D C",
"A3 A3",
"A3 F E D",
"G G G G A G F D",
"C C5 B A G F G",
"E E E G C D",
"E E E G C D",
"E F G A C E D F",
"E C D E F G A G",
"F F F F F F",
"F E E E E E",
"E D D D D E",
"D D E F G F E D",
"E E E G C D",
"E E E G C D",
"E F G A C E D F",
"E C D E F G A G",
"F F F F F F",
"F E E E E E",
"G C5 B A G F E D",
"C C E G C5")
duration <- c(1, 3, 2,
3, 1, 2,
1, 2, 1.5, 0.5,
4,
1, 1, 1, 1,
1, 1, 1, 1,
1, 1, 3, 1,
2, 2,
1, 1, 2,
1, 1, 2,
1, 1, 1.5, 0.5,
4,
1, 1, 1, 1,
1, 1, 1, 0.5, 0.5,
1, 1, 1, 1,
4,
1, 1, 1, 1,
3, .5, .5,
1, 1, 1, 1,
4,
1, 1, 1, 1,
4,
1, 1, 1, 1,
4,
1, 1, 1, 1,
4,
1, 1, 1, 1,
3, 1,
1, 1, 1, 1,
1, 1, 1, 1,
1, 1, 1, 1,
1, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
1, 1, 0.5, 0.5, 0.5, 0.5,
1, 1, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
1, 0.5, 0.5, 1, 0.5, 0.5,
1, 0.5, 0.5, 1, 0.5, 0.5,
1, 0.5, 0.5, 0.5, 0.5, 1,
1, 0.33, 0.33, 0.33, 1, 0.33, 0.33, 0.33,
1, 1, 0.5, 0.5, 0.5, 0.5,
1, 1, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
1, 0.5, 0.5, 1, 0.5, 0.5,
1, 0.5, 0.5, 1, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
1, 0.33, 0.33, 0.33, 2)
jbells <- data_frame(pitch = strsplit(pitch, " ")[[1]],
duration = duration)
jbells <- jbells %>%
mutate(octave = substring(pitch, nchar(pitch)) %>%
{suppressWarnings(as.numeric(.))} %>%
ifelse(is.na(.), 4, .),
note = notes[substr(pitch, 1, 1)],
note = note + grepl("#", pitch) -
grepl("b", pitch) + octave * 12 +
12 * (note < 3),
freq = 2 ^ ((note - 60) / 12) * 440)
tempo <- 250
sample_rate <- 44100
make_sine <- function(freq, duration) {
wave <- sin(seq(0, duration / tempo * 60, 1 / sample_rate) *
freq * 2 * pi)
fade <- seq(0, 1, 50 / sample_rate)
wave * c(fade, rep(1, length(wave) - 2 * length(fade)), rev(fade))
}
jbells_wave <- mapply(make_sine, jbells$freq, jbells$duration) %>%
do.call("c", .)
play(jbells_wave)