Calculus II – Final Exam Review
Fullerton College
1. Use the Trapezoidal Rule and Simpson’s Rule
with n = 4, to approximate the value of the
definite integral.
Z 3
2
dx
1
+
x2
2
10. Find the x-coordinate of the centroid of the
region bound by the graphs of y = ax − x2
and y = 0.
For exercises 11–16, evaluate each integral by
hand.
Z
Z
x2
11.
dx
12.
y 2 ln y dy
x+3
Z ∞
Z
2 −2x
13.
x e
dx
14.
cos3 (πθ) dθ
2. Evaluate the integral.
Z
x
dx
4 + 9x4
3. find f 0 (x) for f (x) = x2 cosh 3x
0
Z
4. Solve the differential equation.
15.
dy
=y+2
dx
Z
16.
5. Determine the area bounded by the graphs of
the following two equations.
x = 2y − y 2 ,
√
x2
dx
25 − x2
4
dt
(t − 1)(t − 2)(t − 3)
17. Evaluate the limit using L’Hôpital’s Rule.
x=y−6
lim xe−x
6. Write the integral for the disk method and
shell method for finding the volume of the solid
generated by revolving the region bounded by
the graphs of the following equations about
the line x = 3. Evaluate one of your integrals
to find the volume.
18. Write an expression for the nth term of the
sequence.
−
y = x2 , x = 2, y = 0
4 5/4
x ,
5
1 3
3 27
,
,− ,
, ...
11 13
5 17
19. Express the repeating decimal 0.05 as a geometric series using sigma notation and use the
sum of the series to rewrite the number as a
fraction of integers.
7. Find the arc length of the graph of the function
over the given interval.
f (x) =
2
x→∞
[0, 9]
For exercises 20–29, determine if each series converges or diverges and justify your answer. Use
each test at least once.
8. Find the surface area of the solid generated
by revolving the the region bounded by the
graphs of the following equations about the
x-axis.
20.
∞ n
X
π
n=0
√
y = 4 x, x = 5, x = 12, y = 0
22.
n
∞ X
3n + 1
n=0
9. A force of 4 pounds is needed to stretch a
spring from its natural length of 8 inches to
a length of 16 inches. Find the work done in
stretching the spring from 1 foot in length to
2 feet.
3
∞
X
4n − 5
21.
∞
X
(−1)n ln (n)
n
n=2
23.
∞
X
n=5
√
1
n−4
∞
X
1
1
(−1)n+1 5n
−
25.
24.
2n − 1 2n + 1
n!
n=1
n=1
1
Calculus II – Final Exam Review
26.
∞
X
1
√
5
n6
27.
1
3
n − 2n2
n=3
29.
n=1
28.
∞
X
ne−n
Fullerton College
36. Find dy/dx and d2 y/dx2 at the point t = 2
for parametric curve:
2
n=1
∞
X
n
∞ X
1
1+
n
n=1
x = 2 − t3 ,
37. Find two sets of polar coordinates with 0 ≤
θ < 2π, that represent the√same point
as the
rectangular coordinate, 2 3, −2 .
30. Find the interval and radius of convergence
for the power series.
38. Convert the rectangular equation to polar.
∞
X
(−1)n xn
n
n=1
y2
x2
−
=1
4
9
31. Find the power series for the function centered
at c = −2
f (x) =
39. Convert the polar equation to rectangular.
1
2x + 3
r = 6 sin(θ)
40. Find all points of vertical and horizontal tangency on the polar curve.
32. Find the Maclaurin series for the function,
2
f (x) = ex .
r = 4 − 4 cos θ
33. Use the Maclaurin series to approximate the
value of the integral with an error of less than
0.00001.
Z
41. Find the area of the interior of the rose curve
1
r = 12 sin(3θ)
sin x3 dx
0
34. Find a set of parametric equations that represent the straight line starting at the point
(−2, 5) and ending at the point (4, 1).
35. Find a set of parametric equations that represent the ellipse shown below.
4
2
4
–4
y = 4t + ln(t)
–2
–4
2
Calculus II – Final Exam Review
Fullerton College
20. Diverges; geometric with |r| =
1. Trapezoidal: 0.2848; Simpson’s: 0.2838
2
1
3x
2.
arctan
12
2
π
3
>1
21. Converges; alternating series test
22. Converges; root test with L =
3
4
<1
23. Diverges; directly compare to
P
√1
n
2
3. 2x cosh 3x + 3x sinh 3x
x
4. y = Ce − 2
24. Converges; telescoping
125
un2
5.
6
Z 4
√
6. disk: π
((3 − y)2 − 1) dy
25. Converges; ratio test with L = 0 < 1
26. Converges; p-series with p =
6
5
>1
0
Z
shell: 2π
2
27. Converges; integral test
(3 − x)x2 dx
0
28. Converges; limit compare to
3
volume: 8π un
7.
n→∞
30. R = 1, (−1, 1]
31.
9. 60 in-lbs or 5 ft-lbs
a
10. x̄ =
2
12.
y3
(3 ln(y) − 1) + C
9
13.
1
4
32.
∞
X
x2n
n!
n=0
33.
449
1920
(− 25 , − 32 )
34. x = 6t − 2,
y = −4t + 5
35. x = 6 cos t,
y = 3 sin t
38. r2 =
36
9 cos2 θ − 4 sin2 θ
39. x2 + (y − 3)2 = 9
17. 0
18. an =
n=0
−2n (x + 2)n ,
19
3
36. − ; −
8
576
11π
5π
37. 4,
, −4,
6
6
1
(3 sin(πθ) − sin3 (πθ)) + C
3π
√
x
1
25 arcsin − x 25 − x2 + C
15.
2
5
2
(t − 1) (t − 3)2
16. ln
+C
(t − 2)4
14.
19.
∞
X
n=0
x2
− 3x + 9 ln(x + 3) + C
2
∞
X
1
n3
29. Diverges; lim an = e 6= 0
232
un
15
592π
un2
8.
3
11.
P
√
√
40. horizontal: (−3,
(−3, −3 3)
√ 3 3),√
vertical: (1, 3), (1, − 3), (−8, 0)
cusp: (0, 0)
(−1)n 3n−1
2n + 9
0.05(0.1)n =
1
18
41. 36π un2
3