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
0t

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 )
n0 2

e) 
n0
2 n 1
(  1) e
n
4

b)

n

5 (3 )
n2 2
c)
2 n 1


1
n2n
2
1
d) 
n4 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