Área real

\[\int_{0}^{2e}ln(x)dx = \int_{0}^{1}ln(x)dx+\int_{1}^{2e}ln(x)dx\]

Primera parte

\[\int_{0}^{1}ln(x)dx = \lim_{b \to 0} \int_{b}^{1}ln(x)dx = \lim_{b \to 0} xlnx-x|_b^1\] \[= [\lim_{b \to 0} 1ln(1)-1] - [\lim_{b \to 0} tln(t)-t]]\]

\[= \lim_{b \to 0} \cancelto{0}{ln(1)} -\lim_{b \to 0}\cancelto{1}{1}- \lim_{b \to 0} \cancelto{0}{tln(t)}+\lim_{b \to 0}\cancelto{0}{t}\]

\[\int_{0}^{1}ln(x)dx = -1\]

Segunda parte

\[\int_{1}^{2e}ln(x)dx = xlnx-x|_1^{2e}\]

\[= [2e*ln(2e)-2e] - [1ln(1)-1] = [3.768339]-[-1] = 4.768339 \]

\[\int_{1}^{2e}ln(x)dx = 4.768339\]

resultado1 <- 2*exp(1)*log(2*exp(1))-2*exp(1)
resultado1
## [1] 3.768339
resultado2 <- log(1)-1
resultado2
## [1] -1

Total

\[\int_{0}^{2e}ln(x)dx = -1 + 4.768339 = 3.768339 \\ \int_{0}^{2e}ln(x)dx = 3.768339\]

Área Simulada

Recta de la función

plot(log, xlim = c(0,5.6))
abline(h = 0, col = 'red', lty = 2)
abline(v = 1, col = 'red', lty = 2)
text(1.5,1, 'ln(x)')
text(0.3, -0.3, 'Primera \n parte')
text(4, 0.6, 'Segunda parte')

Simulación Montecarlo 1

Primera parte

r = 200000
d = c(); dn1 = c(); x = c(); y = c()
for(n in 1:r){
  x[n] = runif(1, 0, 1)
  y[n] = runif(1, -10, 0)
  d[n] = (y[n]>log(x[n]))
  dn1[n] = ifelse(d[n] == T, 'Interno', 'Externo')
}
plot(x, y, col = ifelse(dn1 == 'Interno', 'red', 'black'), pch = 19, cex = 0.2)

proporcion_1 = (length(dn1[dn1 == 'Interno'])/r)
proporcion_1
## [1] 0.10042
areafinal_1<-proporcion_1*(-10*1)
areafinal_1
## [1] -1.0042

Segunda parte

r = 200000
d = c(); dn2 = c(); x = c(); y = c()
for(n in 1:r){
  x[n] = runif(1, 1, (2*exp(1)))
  y[n] = runif(1, 0, 2)
  d[n] = (y[n]<log(x[n]))
  dn2[n] = ifelse(d[n] == T, 'Interno', 'Externo')
}
plot(x, y, col = ifelse(dn2 == 'Interno', 'red', 'black'), pch = 19, cex = 0.2)

proporcion_2 = (length(dn2[dn2 == 'Interno'])/r)

areafinal_2 <- proporcion_2*(2*(2*exp(1)-1))
areafinal_2
## [1] 4.747301

Área total simulada

area_montecarlo <- areafinal_1+areafinal_2
area_montecarlo
## [1] 3.743101

Simulación con trapecios

n=100

Logan <- function(x) log(x)
x1 <-seq(0.001, 5.436564, by = 0.01)

n=100
a=0.001
b=5.436564
h=(b-a)/n
xi<-a+(0:n)*h
xi
##   [1] 0.00100000 0.05535564 0.10971128 0.16406692 0.21842256 0.27277820
##   [7] 0.32713384 0.38148948 0.43584512 0.49020076 0.54455640 0.59891204
##  [13] 0.65326768 0.70762332 0.76197896 0.81633460 0.87069024 0.92504588
##  [19] 0.97940152 1.03375716 1.08811280 1.14246844 1.19682408 1.25117972
##  [25] 1.30553536 1.35989100 1.41424664 1.46860228 1.52295792 1.57731356
##  [31] 1.63166920 1.68602484 1.74038048 1.79473612 1.84909176 1.90344740
##  [37] 1.95780304 2.01215868 2.06651432 2.12086996 2.17522560 2.22958124
##  [43] 2.28393688 2.33829252 2.39264816 2.44700380 2.50135944 2.55571508
##  [49] 2.61007072 2.66442636 2.71878200 2.77313764 2.82749328 2.88184892
##  [55] 2.93620456 2.99056020 3.04491584 3.09927148 3.15362712 3.20798276
##  [61] 3.26233840 3.31669404 3.37104968 3.42540532 3.47976096 3.53411660
##  [67] 3.58847224 3.64282788 3.69718352 3.75153916 3.80589480 3.86025044
##  [73] 3.91460608 3.96896172 4.02331736 4.07767300 4.13202864 4.18638428
##  [79] 4.24073992 4.29509556 4.34945120 4.40380684 4.45816248 4.51251812
##  [85] 4.56687376 4.62122940 4.67558504 4.72994068 4.78429632 4.83865196
##  [91] 4.89300760 4.94736324 5.00171888 5.05607452 5.11043016 5.16478580
##  [97] 5.21914144 5.27349708 5.32785272 5.38220836 5.43656400
plot(x1, Logan(x1), col = 'darkred', type = 'l', lwd = 2, main = 'Trapecio')
for(i in 1:n) {
  segments(xi[i], 0, xi[i+1], 0, col = rgb(0.4,0.2,0.7), lwd = '2')
  segments(xi[i+1], 0, xi[i+1], Logan(xi[i+1]), col = rgb(0.4,0.2,0.7), lwd = '2')
  segments(xi[i+1], Logan(xi[i+1]), xi[i], Logan(xi[i]), col = rgb(0.4,0.2,0.7), lwd = '2')
  
}

pot1<-(h/2)*(Logan(a)+Logan(b))
pot2<-(h/2)*(sum(2*Logan(a+(1:(n-1)*h))))

proba1 <- pot1+pot2
proba1
## [1] 3.713218

n=300

Logan <- function(x) log(x)
x1 <-seq(0.001, 5.436564, by = 0.01)

n=300
a=0.001
b=5.436564
h=(b-a)/n
xi<-a+(0:n)*h
xi
##   [1] 0.00100000 0.01911855 0.03723709 0.05535564 0.07347419 0.09159273
##   [7] 0.10971128 0.12782983 0.14594837 0.16406692 0.18218547 0.20030401
##  [13] 0.21842256 0.23654111 0.25465965 0.27277820 0.29089675 0.30901529
##  [19] 0.32713384 0.34525239 0.36337093 0.38148948 0.39960803 0.41772657
##  [25] 0.43584512 0.45396367 0.47208221 0.49020076 0.50831931 0.52643785
##  [31] 0.54455640 0.56267495 0.58079349 0.59891204 0.61703059 0.63514913
##  [37] 0.65326768 0.67138623 0.68950477 0.70762332 0.72574187 0.74386041
##  [43] 0.76197896 0.78009751 0.79821605 0.81633460 0.83445315 0.85257169
##  [49] 0.87069024 0.88880879 0.90692733 0.92504588 0.94316443 0.96128297
##  [55] 0.97940152 0.99752007 1.01563861 1.03375716 1.05187571 1.06999425
##  [61] 1.08811280 1.10623135 1.12434989 1.14246844 1.16058699 1.17870553
##  [67] 1.19682408 1.21494263 1.23306117 1.25117972 1.26929827 1.28741681
##  [73] 1.30553536 1.32365391 1.34177245 1.35989100 1.37800955 1.39612809
##  [79] 1.41424664 1.43236519 1.45048373 1.46860228 1.48672083 1.50483937
##  [85] 1.52295792 1.54107647 1.55919501 1.57731356 1.59543211 1.61355065
##  [91] 1.63166920 1.64978775 1.66790629 1.68602484 1.70414339 1.72226193
##  [97] 1.74038048 1.75849903 1.77661757 1.79473612 1.81285467 1.83097321
## [103] 1.84909176 1.86721031 1.88532885 1.90344740 1.92156595 1.93968449
## [109] 1.95780304 1.97592159 1.99404013 2.01215868 2.03027723 2.04839577
## [115] 2.06651432 2.08463287 2.10275141 2.12086996 2.13898851 2.15710705
## [121] 2.17522560 2.19334415 2.21146269 2.22958124 2.24769979 2.26581833
## [127] 2.28393688 2.30205543 2.32017397 2.33829252 2.35641107 2.37452961
## [133] 2.39264816 2.41076671 2.42888525 2.44700380 2.46512235 2.48324089
## [139] 2.50135944 2.51947799 2.53759653 2.55571508 2.57383363 2.59195217
## [145] 2.61007072 2.62818927 2.64630781 2.66442636 2.68254491 2.70066345
## [151] 2.71878200 2.73690055 2.75501909 2.77313764 2.79125619 2.80937473
## [157] 2.82749328 2.84561183 2.86373037 2.88184892 2.89996747 2.91808601
## [163] 2.93620456 2.95432311 2.97244165 2.99056020 3.00867875 3.02679729
## [169] 3.04491584 3.06303439 3.08115293 3.09927148 3.11739003 3.13550857
## [175] 3.15362712 3.17174567 3.18986421 3.20798276 3.22610131 3.24421985
## [181] 3.26233840 3.28045695 3.29857549 3.31669404 3.33481259 3.35293113
## [187] 3.37104968 3.38916823 3.40728677 3.42540532 3.44352387 3.46164241
## [193] 3.47976096 3.49787951 3.51599805 3.53411660 3.55223515 3.57035369
## [199] 3.58847224 3.60659079 3.62470933 3.64282788 3.66094643 3.67906497
## [205] 3.69718352 3.71530207 3.73342061 3.75153916 3.76965771 3.78777625
## [211] 3.80589480 3.82401335 3.84213189 3.86025044 3.87836899 3.89648753
## [217] 3.91460608 3.93272463 3.95084317 3.96896172 3.98708027 4.00519881
## [223] 4.02331736 4.04143591 4.05955445 4.07767300 4.09579155 4.11391009
## [229] 4.13202864 4.15014719 4.16826573 4.18638428 4.20450283 4.22262137
## [235] 4.24073992 4.25885847 4.27697701 4.29509556 4.31321411 4.33133265
## [241] 4.34945120 4.36756975 4.38568829 4.40380684 4.42192539 4.44004393
## [247] 4.45816248 4.47628103 4.49439957 4.51251812 4.53063667 4.54875521
## [253] 4.56687376 4.58499231 4.60311085 4.62122940 4.63934795 4.65746649
## [259] 4.67558504 4.69370359 4.71182213 4.72994068 4.74805923 4.76617777
## [265] 4.78429632 4.80241487 4.82053341 4.83865196 4.85677051 4.87488905
## [271] 4.89300760 4.91112615 4.92924469 4.94736324 4.96548179 4.98360033
## [277] 5.00171888 5.01983743 5.03795597 5.05607452 5.07419307 5.09231161
## [283] 5.11043016 5.12854871 5.14666725 5.16478580 5.18290435 5.20102289
## [289] 5.21914144 5.23725999 5.25537853 5.27349708 5.29161563 5.30973417
## [295] 5.32785272 5.34597127 5.36408981 5.38220836 5.40032691 5.41844545
## [301] 5.43656400
plot(x1, Logan(x1), col = 'darkred', type = 'l', lwd = 2, main = 'Trapecio')
for(i in 1:n) {
  segments(xi[i], 0, xi[i+1], 0, col = rgb(0.7,0.2,0.2), lwd = '2')
  segments(xi[i+1], 0, xi[i+1], Logan(xi[i+1]), col = rgb(0.7,0.2,0.2), lwd = '2')
  segments(xi[i+1], Logan(xi[i+1]), xi[i], Logan(xi[i]), col = rgb(0.7,0.2,0.2), lwd = '2')
  
}

pot1<-(h/2)*(Logan(a)+Logan(b))
pot2<-(h/2)*(sum(2*Logan(a+(1:(n-1)*h))))

proba2 <- pot1+pot2
proba2
## [1] 3.763294

Simulación Montecarlo 2

set.seed(5234)
G <- function(x){
  return(log(x))
}

monte_carlo <- function(G, a, b, M){
  s = 0
  for(i in 1:M){
    s = s+G(a+(b-a)*runif(1,0,1))
  }
  return(((b-a)/M)*s)
}

res3 <- monte_carlo(G, 0, 5.436564, 1000000)
res3
## [1] 3.767157

esta simulacion arroja distintos resultados cada ves que se corre ocilando entre 3.74 y 3.77 por lo cual no es precisa. Se definió una semilla para obtener un único valor

Comparación

Área Real

\[A_r = 3.768339\]

Simulación Montecarlo 1

\[A_{m1} = 3.76248\]

Simulación Trapecios

n=100

\[A_{t100} = 3.713218\]

n=300

\[A_{t300} = 3.763294\]

Simulación Montecarlo 2

\[A_{m1} = 3.767157\]