Math 152 Exam 2 Review
1. Find each antiderivative:
a)
b)
x
2
( 4 x 2 ) 3 2 dx
(x
2
4)
x
3 2
dx
3
2. Find each antiderivative.
a)
b)
4
x
2x
( x 1)
2
dx
( x 4)
( x 2) ( x 4)
2
2
dx
3. Determine whether or not the improper integral converges and evaluate it if it
converges.
1
a)
1
x ( x 2 ) dx
1
b)
2
c)
x
2
1
x
2
dx
1
1
dx
ln x
d) For what values of p does
1
x (ln x ) p
dx
converge?
2
4. Find the length of the curve given by
3
x ( t ) cos t
3
y ( t ) sin t
0t
2
.
5. The curve of problem 4 can be written as x 2 3 y 2 3 1 0 x 1 0 y 1 .
Find the surface area of the object formed if this curve is rotated about the x-axis.
Set up the integral with respect to x and again with respect to y and again with respect to
t.
6. Which of the sequences are bounded? which converge?
a)
{sin( 1 n )}
c)
{sin n }
e)
ln n
n
b)
d)
( n 1)
{tan
}
2n
{ln( n 2 n ) ln( 4 n 5 n )}
f)
2
n
2
2
e
n
7. Which of the sequences in 6 above are monotone increasing/decreasing?
8. Determine whether the sequence is monotone increasing, decreasing or neither.
a1 1 , a n 1 2
a)
3
b)
an
a1 1 , a n 1 2
3
an
9. Find the limit of each sequence.
a ) arctan
2
n 1
1
b ) n sin
n
n
10. Find the sum of each series.
a)
n
5 (3 )
n0 2
e)
n0
2 n 1
( 1) e
n
4
b)
n
5 (3 )
n2 2
c)
2 n 1
1
n2n
2
1
d)
n4 n
4
2
4n 3
n
n 1
11. Give the conclusion of the divergence test in each case or state no conclusion can be
made from the divergence test.
1
a ) sin
n 1
n
1
b ) n sin
n 1
n
c ) cos n
n 1
d ) ( 1) e
n 1
n
1 n
0
