# Page 1 ```Math 152, Spring 2008-copyright Joe Kahlig
Page 1
(7 points each) For questions 1-4, sketch the bounded region and then set up the integral(s) that would be
used to compute the volume of the indicated solid. You do not have to compute out the integral.
1. A solid is obtained by rotating the region bounded by the given curves about the x-axis.
√
y = 25 − x2
y=0
x=2
x=4
2. A solid is obtained by rotating the region in the first quadrant bounded by the given curves about the
y-axis.
x = y2
x = 2y
3. A solid has its base in the xy-plane. The base is the triangle region with vertices (0, 0), (0, 4), and (3, 12).
Parallel cross section are perpendicular to the x-axis and have the shape for a square.
4. A solid is obtained by rotating the region bounded by the given curves about x = −1.
y = 4x − x2
y = 8x − 2x2
5. (6 points) Set up the integral(s) that would be used to compute the area of the region in the xy-plane
bounded by the following curves.
x = y2 − 1
√
x= y
y=0
y=1
6. (6 points) Find the area of the region bounded by the following curves on the interval [−2, 1]. (yes, I
actually want you to compute this one out to an answer.)
y = e2x
y =1−x
7. (5 points) A force of 40N is required to maintain a spring stretched from its natural length of 12cm to
17cm. How much work (in joules) is done in stretching the spring from 13cm to 20cm?
8. (7 points) A cable the weighs 3 lbs/ft is used to lift 800 lb of coal up a mine shaft 700 ft deep. Find the
word done.
9. (6 poitns) Find the average value of the function f (x) = x3 ln(x) on the interval [1, 3].
10. (6 points) Write out the partial fraction decomposition. You do not have to solve for the constants.
x+2
2
x(x − 9)(x2 + 5)2
11. (6 points each) Evaluate these integrals.
Z
(a)
x3 cos(x2 ) dx
Z
(b)
sin2 (x) cos2 (x) dx
Z 3
x − 2x2 − 4
dx
(c)
x3 − 2x2
Z
(d)
tan3 (2x) sec2 (2x) dx
Z
p
(e)
x3 1 − 4x2 dx
Z √ 2
x −9
dx
(f)
x
Check the back of the page for more problems.
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