1.

Use integration by substitution to solve the integral below

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.

3.

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

4.

Find the area of the region bounded by the graphs of the given equations

5.

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.

This one involves a very unfriendley derivative of a Riemann sum, so we’ll cheat and use R

inventory_cost <- function(orders){
  time <- 0
  max_index <- orders - 1
  for (i in 0:max_index){
    time <- time + (110 - 110*(1/orders)*i)/orders
  }
  
  return (8.25*orders + 3.75*time)
}

solution <- optimize(inventory_cost, 1:110)
round(solution$minimum,1)
## [1] 5

6.

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

7.

Determine whether f ( x ) is a probability density function on the interval. If not, determine the value of the definite integral. ???

Given that the integral evaluates to 1, this is a proper pdf.