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December 10 Mathematics 106a Mr. Haines
2008 Calculus II Final Examination
(5) I. Write the fraction
2 x x −
as the sum of rational functions.
(5) II. If dx x
∫ p
∞
1
converges to 1/2, what is the value of p? Justify your answer.
(15) III. Find the integrals:
A. ∫ 12 x cos( x ) dx
2 3 .
B. ∫
dx x x
x
( 1 )
2
2 .
C. ∫
/ 4
0
3 2 tan sec
π
x xdx.
(5) V. Starting with the Maclaurin series for 1 + x
A. find a power series expression for ∫
dx x
4 1
B. For what values of x does this formula hold?
(10) VI. A spherical tank of radius 20 feet is buried 8 feet below ground and is filled to a height
of 13 feet from the bottom of the tank with water (62.5 pounds per cubic foot). Write an
integral equal to the work done in pumping all the water to ground level.
(5) VII. Consider the region bounded by y = 0, x = 2, and
2 y = x. Write an integral equal to the
volume of the object created when the region is revolved about the line x = 7.
(10) VIII. Find the solution that passes through (0, 2) for the equations:
A.
y
x
dx
dy cos
B.
x xe dx
dy
(10) XI. Do these integrals converge? Justify your answers.
A. dx x
∞
(^2) + 3 1
B. dx x
x
∞
1 3 /^2
cos
(5) XII. If the series ∑
∞
=
k
k
converges, to what does it converge? If it diverges, explain why.
(5) XIII. Test to determine whether the series ∑
∞
= 1 12 +^5
n n
n converges or diverges and explain why.
(5) XIV. For the series
n
n
n x
∞
=
1
1
2
A. Give its radius of convergence:
B. Give its interval of convergence:
C. Give a value of x for which it diverges:
