Exam# 1        Answers are in red

Directions: Show all of your work. Start each problem on a fresh sheet of paper. Do not write on the backside of the pages.

1. Complete the following sentences.
   1. òf(x) dx = F(x) + C means [d /dx]F(x) = f(x) .
   2. If [d /dx]F(x) = [d /dx]G(x) for all x such that a £ x £ b , then there is a constant C such that F(x) = G(x) + C for all x such that a £ x £ b .
2. Assuming that
   

   ó õ x sin(x) dx = -x cos(x) + ó õ cos(x) dx

   [then]
   

   ó õ p -p  x sin(x) dx

   equals 2p
3. The curves y = x² and y = 3x + 4 intersect at the points (-1, 1) and (4, 16) . Write down the integral or integrals that give the area of the region enclosed by the curves y = x² and y = 3x + 4 .
   Since the curve y = 3x + 4 is above the curve y = x² for -1 £ x £ 4
   

   area = ó õ 4 -1  (3x + 4 - x2) dx
4. Write down an integral for the volume of a pyramid with height  10 whose base is a 3  by  3 square. Carefully explain how you arrived at your integral. Well chosen drawings might be helpful.
   The cross-sections parallel to the base are squares whose sides change linearly from s = 3 when y = 0 to s = 0 when y = 10 . Thus, s = 3 - 3y/10 = 3(10 - y)/10 . Therefore,
   

   volume = 32 102 ó õ 10 0  (10 - y)2dy
5. Use calculus, not calculators, to work out the following integrals. (It's best to check your answers.)
   1. òln(x)  dx = xlnx - x + C
   2. ò2x cos(x²)  dx = sin(x²) + C
   3. òx e^2x  dx = [x/2]e^2x - [1/4]e^2x + C
   4. ò[1/(x² - 4)]  dx = [1/4] ln|x - 2| - [1/4] ln|x + 2| + C

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File translated from T_EX by T_TH, version 3.72.
On 20 Feb 2006, 14:18.