HW 13

1) Integration by Substitution

\[\int 4e^{-7x} dx\]
\[ let\ U = -7x\]
\[dU = -7 dx\]

\[-\frac{4}{7}\int e^{U} dU\] \[-\frac{4}{7}e^U +C\] \[-\frac{4}{7}e^{-7x} +C\]

2) Bacteria Contamenation

Rate of Change:
\[\frac{dN}{dt} = -\frac{3150}{t^4} - 220\] \[N(1) = 6530\]

Solve for N(t):
\[N(t) = \int-\frac{3150}{t^4}dt - \int220dt\]
\[N(t) = -\frac{1}{3}\frac{3150}{t^3} - 220t + N_0\]
\[6530 = -\frac{3150}{3} - 220 + N_0\] \[N_0 = 7800\]
\[N(t) = -\frac{1}{3}\frac{3150}{t^3} - 220t + 7800\]

3) Area Under Line

fun = function(x){
  (2*x) - 9
}
integrate(fun,4,8.5)
## 15.75 with absolute error < 1.8e-13

4) Bounded Region Area

\[x+2 = x^2 - 2x -2\] \[x=-1 \ x=4\]

library(cubature)
## Warning: package 'cubature' was built under R version 3.3.3
fun1 = function(x){
  x+2
}
fun2 = function(x){
  (x^2)- (2*x) -2
}
hcubature(fun1,-1,4)$integral - hcubature(fun2,-1,4)$integral
## [1] 20.83333

5) Minimize Costs

\[x = 110/orders\] \[C=8.25orders + \frac{3.75x}{2}\]
\[C=8.25orders + \frac{206.25}{orders}\] \[C'=8.25 - \frac{206.25}{orders^2}\] Set C’=0:
\[orders=\sqrt{\frac{206.25}{8.25}}\] \[orders=5\]

6)Integration by Parts

\[\int ln(9x)x^6 dx\]
\[U = ln(9x)\\ dV=x^6\]
\[dU = \frac{1}{x}\\ V=\frac{1}{7}x^7\] \[UV - \int{VdU}\]

\[ln(9x)\frac{1}{7}x^7 - \int{\frac{1}{7}x^7\frac{1}{x}}\]
\[\frac{1}{7}(ln(9x) - \int{x^6}\]
\[\frac{1}{7}x^7 |ln(9x) -)\frac{1}{7} | - C\]

7) Probability Density Function

\[\int_{1}^{e^6}{\frac{1}{6x}dx}\] \[\frac{1}{6}(ln(e^6)-ln(1))\]

\[\frac{1}{6}(6-0) = 1\] The function is a probability density function on the interval [1,e^6]