9 FURTHER APPLICATIONS OF INTEGRATION

CHAPTER 9
SAMPLE EXAM
6. Let f (x) = 3√ x. We wish to put upper and lower bounds on the length of f (x) from x = 0tox = 1.
(a) Write an integral that, when evaluated, will give us the length that we want.
(b) Explain why the integral you wrote in part (a) is improper.
(c) Give a geometrical reason why the integral in part (a) converges.
7. An airline notes that if the price of an 8 A.M. round trip ticket from Boston to New York is set at $250,
they can sell 1000 tickets. For every $50 increase in price, they can sell 100 fewer tickets. For every $50
decrease in price, they can sell 100 more, up to their capacity of 2000 seats.
(a) Find the demand function.
(b) If they go ahead and charge $250, what is the total consumer surplus?
(c) What does the quantity you found in part (b) represent?
(d) How many people are willing to pay $500?
* (e) It annoys the airline that there are people willing to pay $500, but only paying $250. If they set the
price at $500, sell all the tickets they can at that price, and then have a “sale”, changing the price of
tickets to $250, would the consumer surplus increase or decrease? Why?
8. Which of the following can be probability density functions? Why or why not?
(a) f (x) =
1
π ( 1 + x 2)
(b) g (x) = 2
1 + e x
{ 12
e x if x < 0
(c) h (x) =
1
2 e−x if x ≥ 0
9. Let R be the region shaded below.
(a) Find the x-coordinate of the centroid of R without integrating. Explain why your answer is correct.
(b) Find the y-coordinate of the centroid of R by evaluating the appropriate integral.
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