Excercise 1
Use integration by substitution to solve the integral below. \[ \int4e^{-7x} dx \] Solution: \[ u = -7x \] \[ du = -7x\] \[ dx = \frac{du}{-7}\] \[ 4\int e^{u} \frac{du}{-7}\] \[ -\frac{4}{7}\int e^{u} du\] \[ -\frac{4}{7} (e^{u}) +c\] \[ -\frac{4}{7} e^{-7x} +c\]
Excercise 2
Biologists are treating a pond contaminated with bacteria. The level of contamination is changing at a rate of \(\frac{dN}{dt} = -\frac{3150}{t^4}-220\) bacteria per cubic centimeter per day, where t is the number of days since treatment began. Fina a function N(t) to estimate the level of contamination if the level after 1 day was 6530 bacteria per cubic centimeter.
library(mosaic)
library(mosaicCalc)antiD((-3150)*t^(-4)-220~t)## function (t, C = 0)
## 1050 * t^-3 - 220 * t + C\[N(t) = \frac{1050}{t^{-3}}-220t+6530\]
Excercise 3
Find the total area of the red rectangles in the figure below, where the equation of the line is f ( x ) = 2x - 9
fx <- function(x) {2*x-9}
integrate(fx,4.5,8.5)## 16 with absolute error < 1.8e-13Excercise 4
Find the area of the region bounded by the graphs of the given equations. \[y = x^{2} - 2x - 2, y = x + 2\] Enter your answer below.
\[x^{2}-3x-4=0\]
Solving the quadratic equation
a <- 1
b <- -3
c <- -4
(-b+sqrt(b^2-4*a*c))/(2*a)## [1] 4(-b-sqrt(b^2-4*a*c))/(2*a)## [1] -1Based on the plot below we see that the line function will be the upper bound
y1 <- function(x){x^2 - 2*x - 2}
y2 <- function(x){x+2}
curve(expr = y1, from = -5, to=5)
curve(expr = y2, from = -5, to=5, col=2, add=TRUE)
Final Solution: Area bounded between both regions.
fx <- function(x) {x+2-(x^2 - 2*x - 2)}
integrate(fx,-1,4)## 20.83333 with absolute error < 2.3e-13Excercise 6
Use integration by parts to solve the integral below. \[ \int ln(9x) \cdot x^{6} dx \] Solution:
\[u = ln (9x)\] \[v = x^{6}\] \[u' = \frac{1}{dx}ln(9x) = \frac{1}{x}\] \[\int x^{6} = \frac{x^{7}}{7}\] \[ln(9x) \cdot \frac{x^{7}}{7}- \int \frac{1}{x} \cdot \frac{x^{7}}{7}dx\] \[ln(9x) \cdot \frac{x^{7}}{7}- \frac{1}{7}\int {x^{6}}dx\] \[ln(9x) \cdot \frac{x^{7}}{7}- \frac{1}{7}\cdot \frac{x^7}{7}\] \[\frac{x^{7}}{7}(ln(9x) - \frac{1}{7})\]
Excercise 7
Determine whether f ( x ) is a probability density function on the interval 1, \(e^6\) . If not, determine the value of the definite integral.
y1 <- function(x){1/(x*6)}
curve(expr = y1, from = 1, to=exp(6))
integrate(y1,1,exp(6))## 1 with absolute error < 9.3e-05