King Fahd University of Petroleum and Minerals
Department of Mathematics
Math 102
Exam II
231
November 07, 2023
Net Time Allowed: 120 Minutes
MASTER VERSION
231, Math 102, Exam II
Example 3 / Section 7.3
Page 1 of 9
MASTER
1. The volume of the solid formed by revolving the region bounded by the graphs of
y = x2 + 1, y = 0, x = 0 and x = 1 about the y-axis is:
(Hint: The Shell Method is preferable in this question)
3π
2
5π
(b)
2
π
(c)
2
7π
(d)
2
9π
(e)
2
(a)
(correct)
Q33 / Section 8.1
Z 1
2
0
e−x + 1
2.
dx =
e+1
(a) 2 ln
2
e+1
(b) ln
2
e−1
(c) 2 ln
2
e−1
(d) ln
2
e+2
(e) 2 ln
2
(correct)
231, Math 102, Exam II
Page 2 of 9
MASTER
Q31/ Section 8.2
Z
3.
arctan x dx =
1
ln(1 + x2 ) + c
2
1
(b) x arctan x + ln(1 + x2 ) + c
2
1
(c) x2 arctan x − ln(1 + x2 ) + c
2
1
(d) x arctan x − ln(1 + x) + c
2
1
(e) x arctan x − ln(x2 ) + c
2
(a) x arctan x −
(correct)
Example 1 / Section 8.3
Z
4. If
1
1
sin3 x cos4 x dx = − (cos x)m + (cos x)n + c, then m + n =
5
7
(a) 12
(b) 14
(c) 13
(d) 11
(e) 10
(correct)
231, Math 102, Exam II
Q46 /
Page 3 of 9
MASTER
Section 8.3
Z π
2
5.
sin 8x cos 7x dx =
0
(a)
8
15
(b) −
(c)
8
15
3
5
(d) −
(e)
(correct)
3
15
2
3
Example 7/ Section 7.4
6. The√area of the surface formed by revolving the graph of f (x) = x2 on the interval
[0, 2] about the y-axis is:
13π
3
14π
(b)
3
16π
(c)
3
11π
(d)
3
10π
(e)
3
(a)
(correct)
231, Math 102, Exam II
Q22 / Section 8.2
Z
7.
Page 4 of 9
MASTER
ln x
dx =
x3
− ln x
1
−
+c
2x2
4x2
ln x
1
(b)
+
+c
2x2 4x2
− ln x
1
(c)
−
+c
2x2
2x2
1
− ln x
−
+c
(d)
x2
4x2
− ln x
1
(e)
+
+c
2x2
4x2
(a)
(correct)
Q7 / Section 8.4
Z
8.
√
dx
=
x2 − 25
p
x2 − 25| + c
p
(b) ln |x − x2 − 25| + c
x p 2
(c) ln + x − 25 + c
5
x p
(d) ln − x2 − 25 + c
5
√
x2 − 25
(e) ln x −
+c
5
(a) ln |x +
(correct)
231, Math 102, Exam II
Page 5 of 9
MASTER
Example 2 / Section 8.5
Z
9.
5x2 + 20x + 6
dx =
x3 + 2x2 + x
9
x6
−
+c
(a) ln
x+1
x+1
9
x5
−
+c
(b) ln
x+1
x+1
x4
8
(c) ln
−
+c
x+1
x+1
8
+c
(d) 6 ln |x| −
(x + 1)2
x6
8
−
(e) ln
+c
x+1
(x + 1)2
(correct)
Q16 / Section 8.5
10. If
6x
A
Bx + C
=
+
, then A + B + C =
x3 − 8 x − 2 x2 + 2x + 4
(a) 2
(b) 3
(c) 1
(d) −1
(e) 0
(correct)
231, Math 102, Exam II
Page 6 of 9
MASTER
Q57 / Section 8.7
Z π
2
11.
0
dθ
=
1 + sin θ + cos θ
(a) ln 2
(correct)
(b) ln 3
(c) 2 ln 2
(d) 2 ln 3
(e) ln 5
Example 5 / Section 8.3
Z
12. If
tan4 3x tan6 3x
+
+ C, then A + B =
sec 3x tan 3x dx =
A
B
(a) 30
(b) 6
(c) 28
(d) 26
(e) 32
4
3
(correct)
231, Math 102, Exam II
Q17/ Section 7.2
Page 7 of 9
MASTER
13. The volume of the solid generated by revolving the region bounded by the graphs
of the equations y = x, y = 3 and x = 0 about the line y = 4 is
(a) 18 π
(correct)
(b) 17 π
(c) 16 π
(d) 19 π
(e) 20 π
Q73 / Section 7.2
14. The base of a solid is the region bounded by the graphs of y = x + 1 and y = x2 − 1.
If the cross sections of the solid perpendicular to the x-axis are squares, then the
volume of the solid is equal to:
Z 2
(a)
−1
Z 1
(b)
(x4 − 2x3 − 3x2 + 4x + 4) dx
(x4 − 2x3 − 3x2 + 4x + 4) dx
−2
Z 2
(c)
(x4 + 2x3 + 3x2 + 4x + 4) dx
−1
Z −2
(d)
(x4 + 2x3 + 3x2 + 4x + 4) dx
−1
Z 2
(e)
−1
(x4 − 2x3 − 3x2 − 4x + 4) dx
(correct)
231, Math 102, Exam II
Page 8 of 9
MASTER
Example 2 / Section 7.4
1
x3
1
+
on the interval
, 2 is:
15. The arc length of the graph of y =
6
2x
2
33
16
31
(b)
16
29
(c)
16
27
(d)
16
35
(e)
16
(a)
(correct)
Q63 / Section 8.2
Z
16. If
√
√
√
√
sin x dx = a x cos x + b sin x + c, then a + b =
(a) 0
(b) 4
(c) 1
(d) −1
(e) −4
(correct)
231, Math 102, Exam II
Page 9 of 9
MASTER
Q33/ZSection 8.4
√
17.
x
dx =
4x − x2
√
x−2
− 4x − x2 + c
2
√
− 4x − x2 + c
(b) arcsin x−2
2
√
(c) arccos x−2
− 4x − x2 + c
2
√
(d) 2 arcsin x−2
+ 4x − x2 + c
2
√
(e) 2 arccos x−2
− 4x − x2 + c
2
(a) 2 arcsin
(correct)
Q27 / Section 8.5
Z
18.
sec2 x
dx =
tan2 x + 5 tan x + 6
tan x + 2
+c
tan x + 3
tan x − 2
(b) ln
+c
tan x + 3
tan x + 2
(c) ln
+c
tan x − 3
tan x − 2
(d) ln
+c
tan x − 3
tan x + 2
(e)
+c
tan x + 3
(a) ln
(correct)
Math 102, 231, Exam II
Q
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Answer KEY
MASTER CODE01 CODE02 CODE03 CODE04
A
D 4
A 3
C 1
E 3
A
D 5
B 4
A 3
B 2
A
A 3
C 5
A 5
C 5
A
C 2
B 2
A 2
A 1
A
E 1
E 1
C 4
A 4
A
B 10
A 12
B 10
A 12
A
A 11
A 6
E 9
A 7
A
C 6
B 8
B 11
D 6
A
A 9
A 11
E 6
E 10
A
C 12
E 10
A 7
B 11
A
D 7
D 9
B 12
E 9
A
B 8
A 7
E 8
D 8
A
C 16
E 14
C 18
D 16
A
E 15
B 13
B 13
D 17
A
D 17
C 18
B 14
B 14
A
C 14
E 15
B 17
D 15
A
E 18
D 16
E 16
E 13
A
E 13
D 17
E 15
C 18