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Math 152, Summer 2008-copyright Joe Kahlig
1. (5 points) The region bounded by the curves x = 7 − 8y − y 2 and x = 0 on the interval y = 1 to y = 7 is shown
in the picture. Set up the integral(s) that would compute the volume when the region is rotated about the
x−axis.
7
6
5
y 4
3
2
1
–2
0
2
4
6
8
x
2. (9 points) Set up only. Give the integral that will compute the area of the surface when the curve y = 3x5 − 5x4
from the point (1, −2) to the point (3, 324) is rotated about x = −1. You do not need to compute or simplify
the integral.
3. (9 points) Find the arc length of the curve x =
t2
1
(2t + 3)3/2 , y =
+ t on the interval 0 ≤ t ≤ 3
3
2
4. (10 points) Write out the partial fraction decomposition. Solve for the constants.
11x4 + 16x2 + 2x + 5
x(x2 + 1)2
5. (8 points) You have been asked to compute the surface area of a small lake. The table gives the width of the
lake measured at every 20 ft from one end of the lake. Use Simpson’s Rule with n = 4 to estimate the surface
area of the lake. Note: x is the distance from the end of the lake where the measurements were started.
x(ft)
0 20 40 60 80 100 120 140 160
width(ft) 10 25 50 80 100 150 120 80
60
6. (8 points each) Compute these integrals.
Z
1
(a)
dx
(1 + 4x2 )2
Z 3
x + 2x2 + x + 4
dx
(b)
x2 + 1
7. (9 points) Find the function y that will solve the initial value problem.
xy ′ − 2y = x4 ex where x > 0 and y(2) = 16e2
8. (9 points) Find the function y that will solve the differential equation.
y ′ = 4y 2 + 4 + x2 y 2 + x2
9. (9 points) A tank contains 1000L of water that has an initial salt concentration of 0.25 kg/L. Brine [a salt
solution] is pumped into the tank through two pipes. The first pipe pumps a solution at a rate of 5 L/min that
contains a salt concentration of 0.05kg/L. The second pipe pumps a solution at a rate of 15 L/min that contains
a salt concentration of 0.01kg/L. Assume that the solution is well mixed and drains from the tank at a rate of
20 L/min.
Find a formula that gives how much salt is in the tank after x minutes.
10. (7 points) Use the Comparison Theorem to determine whether the integral
Z∞
1
dx is convergent or diverx + e2x
1
gent. Be sure to fully justify your steps.
11. (9 points) Determine whether the integral, given below, is convergent or divergent. If it converges, determine its
value. If it diverges, explain in detail why this is so.
Z10
1
dx
(x − 9)2
0
Check the back of the page for more problems.