Assignment 13

1.Use integration by substitution to solve the integral below

knitr::include_graphics('13_1.jpg')

2.Biologists are treating a pond contaminated with bacteria. The level of contamination is changing at a rate of dN dt =  3150 t 4  220 bacteria per cubic centimeter per day, where t is the number of days since treatment began. Find a function N( t ) to estimate the level of contamination if the level after 1 day was 6530 bacteria per cubic centimeter.

knitr::include_graphics('13_2.jpg')

3.

Find the total area of the red rectangles in the figure below, where the equation of the line is f ( x ) = 2x - 9.

prob_func<- function(x){
    2*x-9
}
area <- integrate(prob_func, lower = 4.5, upper=8.5)
area
## 16 with absolute error < 1.8e-13
  1. Find the area of the region bounded by the graphs of the given equations. y = x2 - 2x - 2, y = x + 2 Finding the zeroes:
knitr::include_graphics('13_4.jpg')

integrate_4 <- function(x) { x + 2 - (x**2 - 2*x - 2)}

area <- integrate(integrate_4, -1, 4)

area
## 20.83333 with absolute error < 2.3e-13

A beauty supply store expects to sell 110 flat irons during the next year. It costs $3.75 to store one flat iron for one year. There is a fixed cost of $8.25 for each order. Find the lot size and the number of orders per year that will minimize inventory costs.

knitr::include_graphics('13_5.jpg')

6.Use integration by parts to solve the integral below.

ln( 9x ) · x 6 dx

knitr::include_graphics('13_6.jpg')

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. f ( x ) = 1/6x

knitr::include_graphics('13_7.jpg')