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Math 191: Calculus II
1. Use the washer method. Find the volume generated by rotating the given region
about x = -1.
The given region: y  x , y = 1, and x = 0.
2. Use the shell method. Find the volume of the solid obtained by rotating the region
bounded by y  4 x  x 2 and y = 3 about the line x = 1.
3. Find the average value of the function f ( x)  x 2 1  x 3 on [0, 2] .
4. Integrate:
 x tan
2
5. Integrate:

1
( x) dx .
ln x 2 dx
x3
1
6. The base of S is the triangular region with vertices (0,0), (1,0) and (0,1). Crosssection perpendicular to the x- axis are equilateral triangles. Find the volume.
7. Integrate:
x
2
cos( 2 x)dx
8. Integrate:  cos 4 ( x) dx
9. Find the area enclosed by the line y  x  1 and the parabola y 2  3x  7 .
10. Find the area of the region enclosed by the parabolas y   x 2  2 and
y  x 2  3x. Sketch the area and show your work clearly.
11. Find the volume of the solid obtained by rotating the region bounded by y  x ,
y
x , about the line x = 2. (Sketch the graph.)
12. Find the volume of the solid obtained by rotating the region bounded by
y  2 x  x 2 , and y = 0 about the line x = 3. (Sketch the graph.)
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13. Evaluate the integral.

4
ln y
y
dy .

14. Evaluate the integral.
 cos
4
x sin 2 x dx
0
15. Evaluate the integral.

x
2  4x  x2
dx
2
e2 x
 e2 x  4e x  3 dx
17. Use the Trapezoidal Rule to approximate the given integral with the specified
value of n. (round your answer to six decimal places.)
16. Evaluate the integral.
2
ln x
 1  x dx,
n = 10.
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18. Evaluate the integral.
x
1
dx .
x2  9
3x 2  x  12
19. Evaluate the integral. 
dx .
x 3  6x
2
20. Determine whether the series is convergent or divergent.

1  2n
If it is convergent, find its sum.

n
n 1 3

21.
1
 (n  1)(n  2)
is convergent, find its sum.
n 1
22. Show that the sequence
an 
n
is increasing or decreasing .
n 1
2
23. Is the sequence an  n  n 2  4n convergent or divergent?
If convergent, find the limit.
24. Find a formula for the sequence.
3 4 5 6 7
{ ,
, ,
,
,......}
4 16 64 256 1024
25. Determine the integral is convergent or divergent. Evaluate it if it is convergent.

1
 xln x 
3
dx
e
26. Evaluate the indefinite integral using trigonometric substitution.

16  x 2 dx
3
27. Evaluate the indefinite integral using partial fraction method.
 x
2
 2x  4
dx .
2
 1  x  1


k  1!3

k 1 k!2 k !
ln k
k
28. Use the integral test to show that the series
k 1
2
converges.

29. Test whether the series
converges or diverges.
n2

 11n 
30. Test whether the series  
 converges or diverges.
n 1  11n  7 
31. Find the Taylor series representation for the function f ( x)  ln( x 2 ) centered at

x=3. (Use summation notation , i.e.,
a x
k 0
k
k
, to present your answer.)
32. Find the first three terms of the Maclaurin series of the function
3/ 4
f ( x)  81  x  .

1
 1  x  x 2  ...   x n when x  1 .
33. Given that
1 x
n0
a. Use the above formula to derive a power series centered at the origin for
1
f ( x) 
. (Write the answers in both expanded form,
1  x2
f ( x)  c0  c1 x  c2 x 2  ...., and using summation notation,

a x
n 0
n
n
.)
b. Integrate both sides to derive a formula for g ( x)  tan 1 x.

34. Determine whether the series

n 1
3n 2  2n
4  n5
converges or diverges.

n2
.
n3  1
n 1
36. Find the radius of convergence and interval of convergence of the series

 1n x n .

2n  3
n 0
35. Use the alternating series test to determine whether the series
n
  1
37. Use the integral test to determine whether the series is convergent or divergent.
First, we should check if the function satisfies all the conditions of the integral

e1 / n
test.

2
n 1 n
4
32 n
n4
n 1
39. Determine whether the series is convergent or divergent.

n3

2
n 1 2n  n  1

38. Is the series convergent or divergent?

2

 1
40. Determine whether the series is convergent or divergent.  1   e  n
n
n 1 

cos( n )
41. Determine whether the series is convergent or divergent. 
n3/ 4
n 1
42. Determine whether the series is absolutely convergent, conditionally convergent

(1) n e1 / n
or divergent.

n3
n 1
43. Verify that y  sin x cos x  cos x is a solution of the initial-valued problem
y '  (tan x) y  cos2 x , y(0) = -1 on the interval 

2
x

2
.
dy
tet
.

dt y 1  y 2
du
 2  2u  t  tu .
45. Solve the differential equation
dt
46. Find the solution of the differential equation that satisfies the given initial
condition.
y(0) = 0.
x cos x  2 y  e3 y y ' ,
44. Solve the differential equation.
y
47. Solve the differential equation xy'  y  xe x by making the change of variable
y
v .
x
x
48. Find the orthogonal trajectories of the family of curves. y 
1  kx
49. One model for the spread of a rumor is that the rate of spread is proportional to
the product of the fraction y of the population who have heard the rumor and the
fraction who have not heard the rumor.
(a) Write a differential equation that is satisfied by y.
(b) Solve the differential equation.
(c) A small town has 1000 inhabitants, At 8 AM, 80 people have
heard a rumor. By noon half the town has heard it. At what time
will 90% of the population have heard the rumor?
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50. Set up, but do not evaluate, an integral for the length of the curve.
x2 y2

1
a 2 b2
51. Find the length of the curve. y  ln(cos x) , 0  x   / 3.
52. Find the arc length function for the curve y  sin 1 x  1  x 2 with starting point
(0,1).
53. Find the area of the surface obtained by rotating the curve about the x-axis.
3/ 2
1
x  y 2  2 , 1  y  2 .
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54. Find the area of the surface obtained by rotating the curve about the x-axis.
y  c  a cosh( x / a) , 0  x  a .
55. Find the area of the surface obtained by rotating the curve about the y-axis.
y  3 x , 1 y  2 .
56. The marginal cost function C ' ( x) was defined to be the derivative of the cost
function. If the marginal cost of manufacturing x meters of a fabric is
C ' ( x)  5  .008x  .000009 x 2 (measured in dollars per meter) and the fixed
start-up cost if C(0)=$20,000, use the Net Change Theorem to find the cost of
producing the first 2000 units.
57. A demand curve is given by p 
450
x8
. Find the consumer surplus when the
selling price is $10.
58. If a supply curve is modeled by the equation p  200  .2 x3 / 2 , find the producer
surplus when the selling price is $400.