Final Exam Review Problems Math 152
J. Lewis
1. Find the volume of the solid formed when the region with the given boundaries is
rotated about the given axis.
y  ln x ,
a)
y  0,
y  ln x ,
b)
c)
y  x
3 2
d)
y  x
3 2
1  x  2 is rotated
y  0,
1  x  2 is rotated
about the
about the
y  axis
line x  3
,
y  0,
x  4 is rotated about the
x - axis
,
y  0,
x  4 is rotated about the
line y  8
2. Find the arc length of
f ( x )  ln(sec
x)
0  x   4
.
3. Find the surface area of the torus formed when the circle centered at (R,0) of radius r is
rotated about the y-axis. Describe this circle as x=R + r cos(t) y=rsin(t).
4. Find the surface area of the surface formed when
the y-axis.
y  x
3 2
0 x  4
is rotated about
5. Find the surface area of the surface formed when
the x-axis.
y  2
x
1 x  3
is rotated about
6. Test each series for convergence.


a)
n


n2

 3n  1
3
n
n2
d)
2
 n
n
n
2


e)
 n
n 1
 4n  5
n
n 1
1
5 4

b)
3
 2
sin( 1 n )
n
7. For what values of p do the series converge?

a) 
n2
ln n
n
p

b) 
n2
1
n
p
ln n

c)

n 1
1
n
3 2
 n
1 2
8. Test each series for convergence. In a and b, also estimate the remainder.


n
a) 
n 1
e

b)
n
n2


e)
n2

ln n
n
3 2

n ln n

c)
n
n2
2
1
n ln n

d)
n2
n
2
 n
n!
3  5  7  ...  ( 2 n  1 )
9. Does the series converge? converge absolutely?

sin n

n 1
n
p 1
p
10. Determine whether the series converges and if it does, estimate
absolutely?

a)

n2
(  1)

n
b)
ln n

Rn
. Does it converge

n
(  1 ) sin( 1 n )
c)

n 1
n
(  1 ) ln( 1  1 n )
n 1
11. Find the Maclaurin series for and find the interval of convergence.
a)
f (x) 
x
2
b)
3  2x
x
f (x) 
(1  x )
2
12. Find the Taylor polynomial
a) of degree 3 for
f ( x )  tan x about
a 

4
b) of degree 2 for
f ( x )  sec
2
a 
x about

4
c) of degree 4 for
f ( x )  cos x about
a 

and find an upper bound on the error
6
| cos x  T 4 ( x ) |
on the interval [0,

].
3
13. a) Find the area of the parallelogram with adjacent edges OA and OB where O is the
origin and A=(-1, 2, 2) and B=(-3, 1, 4).
b) Find the cosine of the angle between OA and OB.
14. Integration by parts
a)
e
b)

x
sin x dx
ln x
dx
x
c)
x
d)
x
2
e
2x
dx
x  2 dx
15. Trigonometric Integrals
a)  sin
2
3
x cos
x dx
dx
b) 
1  sin
c).  sin
2
d)  sec
4
x
2
x cos
x tan
3
x dx
x dx
 4
e)  sec
x tan
3
x dx
0
f)  sin
4 x cos 3 x dx
16. Trigonometric Substitution
a) 
b) 
c) 
x
2
2
(4  x )
x
dx
3 2
2
2
(9  x )
dx
3 2
1
2
(8  4 x  x )
3 2
dx
d)
1
 (x2
e) 
 4)
2
dx
1
x
2
x
2
dx
 16
17. Partial Fractions
a) 
b) 
c) 
x
(x
2
 3 x  16
2
 9 )( x  1 )
2
3x
x
x
x
3
 11 x  16
 4x
2
 4x
2
 3 x  10
2
 2 x  10
dx
dx
dx
18. Work
a) A rope is of uniform density, is 30 ft. long and weighs 50 lbs. A 100 lb weight is
attached at the end. the rope hangs freely over the edge of a tall building. Find the work
done in pulling 12 feet of rope to the top of the building.
b) A spring has a natural length of 2 m. A force of 3N is required to hold the spring at a
length of 2.2m. find the work done in stretching the spring from 2.5 m to 4 m.
c) A tank is in the shape of a half cylinder. The tank is 8m long and the face radius is 2m
the spout is 0.5 m above the top of the tank. If the tank is full of water, find the work
done in pumping all the water out the spout.