Math 172
Exam I
Last Name(print):
February 15, 2001
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You must show all appropriate work to receive full credit.
Work all problems on the paper provided. Turn in your exam with your work.
Do not work on the back on any page. SCHOLASTIC DISHONESTY WILL NOT BE TOLERATED.
Integrals
that might be helpful.
Z
Z
sec xdx = ln | sec x + tan x| + C
csc xdx = ln | csc x − cot x| + C
1
1
sec3 xdx = sec x tan x + ln | sec x + tan x| + C
2
2
Z
−1
1
3
csc xdx =
csc x cot x + ln | csc x − cot x| + C
2
2
Z
1. (10 points each) Do any two of these problems.
(a) Graph the region that is bounded by y = 4x − x2 and y = 8x − 2x2 . Set up the integral(s)
for the volume of the solid generated by rotating the region about x = −1.
(b) Find the exact volume of the solid S whose base is the parabolic region {(x, y)|x2 ≤ y ≤ 1}.
Cross sections of the solid perpendicular to the y-axis are semicircles.
(c) Graph the region that is between y = 0 and y = 2 and is also bounded by x = 0 and
x = 3y − y 2 . Set up the integral(s) for the volume of the solid generated by rotating this
region about the y-axis.
2. (10 points) Write out the partial fraction decomposition. You do not need to solve for the constants.
x+2
(x − 5)3 (x2 + 4)2
3. (10 points) Find the average value of the function f (x) = x cos(x) on the interval [0, π]. give the
exact value.
4. (10 points) Complete the square: 12x − 3x2 + 5
5. (10 points each) Integrate the folowing.
Z
(a)
tan3 (3x) sec3 (3x)dx =
Z
x2 − 3x + 7
dx =
x2 − 4x − 5
Z p
(c)
9 − 4x2 dx
(b)
Z
(d)
x2 ln(x)dx =
Z
(e)
(4x2
1
dx =
− 25)3/2