2414-2-review
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the area of the shaded region.
f(x) = x3 + x2 - 6x
1)
30
25
20
15
10
5
-5 -4 -3 -2 -1-5
-10
-15
-20
(-4, -24) -25
-30
A)
y
g(x) = 6x
(3, 18)
(0, 0)
1 2
3
4
937
12
y = 2x2 + x - 6
5
4
3
2
1
-4
-3
A)
8
3
5
B)
2)
-2
-1 -1
-2
-3
-4
-5
-6
-7
-8
1)
x
768
12
C) 81
12
D)
343
12
y = x2 - 4
2)
y
1
2
3
4
B)
x
11
6
C)
1
9
2
D)
19
3
3)
3)
y
1
1
2
x
y = - x4
-1
-2
y = x2 - 2x
A)
76
15
B)
22
15
C)
7
15
D) 2
Find the area enclosed by the given curves.
4) y = 2x - x2 , y = 2x - 4
4)
37
A)
3
31
B)
3
32
C)
3
34
D)
3
5) y = x, y = x2
1
A)
2
1
B)
6
1
C)
12
1
D)
3
5)
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.
6)
6) y = x, y = 0, x = 0, x = 9
81
9
B) 9π
C)
π
D) 27π
A) π
2
2
7)
7) y = x + 2, y = 0, x = -2, x = 2
64
π
B) 6π
A)
3
8) y = sin 4x, y = 0, 0 ≤ x ≤ A)
1
π
2
9) y = - 5x + 10, y = 5x, x = 0
A) 25π
10) y = x2 , y = 16, x = 0
1024
A)
π
5
C) 2π
D) 16π
π
4
8)
B) 4π
C) 8π
D) 2π
B) 150π
C) 50π
D) 10π
128
B)
π
3
6144
C)
π
5
4096
D)
π
5
9)
10)
2
11) y = x2 + 1, y = 3x + 1
333
π
A)
5
11)
207
C)
π
5
B) 27π
63
D)
π
2
Find the volume of the solid generated by revolving the region about the y -axis.
y2
12) The region enclosed by x = , x = 0, y = - 4, y = 4
4
A)
2048
π
5
B)
128
π
5
C)
32
π
3
12)
D)
64
π
5
3
13) The region enclosed by x = , x = 0, y = 1, y = 6
y
A)
15
π
2
B)
13)
5
π
2
C)
5
π
4
D)
21
π
2
Find the volume of the solid generated by revolving the region about the given line.
14) The region bounded above by the line y = 4, below by the curve y = 4 - x2 , and on the right by the
line x = 2, about the line y = 4
224
256
π
B)
π
A)
15
15
C)
8
π
3
D)
14)
32
π
5
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis.
15)
15) About the x-axis
y
6
5
x = y2 /5
4
3
2
1
1
A)
125
π
4
2
3
4
5
B)
6
x
125
π
2
C)
3
125
π
3
D) 50π
16)
16) About the x-axis
y = 5
5
x = 5
y
4
3
x = 25 - y2
2
1
1
A)
2
3
4
250
π
3
5
x
125
π
3
B)
C)
125
π
6
D) 125π
17)
17) About the y-axis
x = 3
y
4
3
y = 3 - x2 /9
2
1
1
2
3
A) 27π
4
5
x
B) 18π
C)
45
π
2
D)
45
π
4
18) About the y-axis
18)
y
4
3
2
y =3sin(x2 )
1
1.8
A) 3π
x
B) 12π
C) 6π
4
D) 9π
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and
lines about the y-axis.
19)
19) y = 7x3 , y = 7x, for x ≥ 0
A)
28
π
15
B)
7
π
15
C)
14
π
5
D)
14
π
15
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and
lines about the x-axis.
20)
20) x = 4y - y2 , x = 0
A)
32
π
3
B)
64
π
3
C)
128
π
3
D)
256
π
3
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves
about the given lines.
21)
21) y = 3x,
y = x2 ; revolve about the y-axis
A) - 27
π
2
22) y = 4 - x2 ,
32
π
A)
5
B)
y = 4,
27
π
4
C)
189
π
2
x = 2; revolve about the line y = 4
8
256
B) π
C)
π
3
15
D)
27
π
2
22)
224
D)
π
15
Solve the problem.
23) A bead is formed from a sphere of radius 2 by drilling through a diameter of the sphere with a
drill bit of radius 1. Find the volume of the bead.
10
32
5
8
B)
π
C)
π
D) π
A) π
3
3
3
3
Find the length of the curve.
1
1
24) y = x3 + from x = 1 to x = 5
6
2x
A)
25) x = 316
15
B)
24)
79
5
C)
127
6
D)
632
15
y4
1
+ from y = 1 to y = 3
8
4y2
A)
41
4
B)
23)
25)
367
36
C)
5
184
9
D)
92
9
Set up an integral for the length of the curve.
1
1
26) y = 1 - x5 , - ≤ x ≤ 4
4
A)
∫
1/4
-1/4
1/4
C)
∫
-1/4
26)
4 + 25x8
dx
4
B)
4 - 4x5 + 5x4
dx
4(1 - x5 )
D)
-1/4
1/4
4 - 4x5 + 25x8
dx
4(1 - x5 )
5 - 4x5
dx
4(1 - x5 )
27)
∫
B)
-π
0
∫
∫
-1/4
27) x = sin 6y, - π ≤ y ≤ 0
0
1 + 36 cos2 6y dy
A)
C)
∫
1/4
∫
0
-π
0
1 + 6 cos 6y dy
D)
-π
∫
1 + cos2 6y dy
1 + 36 sin2 6y dy
-π
Solve the problem.
28) The gravitational force (in lb) of attraction between two objects is given by F = k/x2 , where x is the
distance between the objects. If the objects are 5 ft apart, find the work required to separate them
until they are 50 ft apart. Express the result in terms of k.
1
1
9
1
k
B)
k
C)
k
D)
k
A)
10
250
50
45
28)
29) A vertical right circular cylindrical tank measures 20 ft high and 8 ft in diameter. It is full of oil
weighing 60 lb/ft 3 . How much work does it take to pump the oil to the level of the top of the
29)
tank? Give your answer to the nearest ft · lb.
B) 1,206,372 ft · lb
A) 10,053 ft · lb
C) 603,186 ft · lb
D) 2,412,743 ft · lb
30) A swimming pool has a rectangular base 10 ft long and 20 ft wide. The sides are 4 ft high, and the
pool is full of water. How much work will it take to lower the water level 2 feet by pumping the
water out over the top of the pool? Assume that the water weighs 62.4 lb/ft 3 . Give your answer to
the nearest ft · lb.
A) 99,840 ft · lb
B) 24,960 ft · lb
C) 12,480 ft · lb
30)
D) 199,680 ft · lb
31) The spring of a spring balance is 6.0 in. long when there is no weight on the balance, and it is 7.5 in.
long with 4.0 lb hung from the balance. How much work is done in stretching it from 6.0 in. to a
length of 10.5 in.?
A) 99 lb·in.
B) 27 lb·in.
C) 3.8 lb·in.
D) 6.0 lb·in.
31)
32) A force of 1100 lb compresses a spring from its natural length of 19 in. to a length of 13 in. How
much work is done in compressing it from 13 in. to 7 in.?
A) 0.29 lb·in.
B) 9900 lb·in.
C) 3300 lb·in.
D) 20,000 lb·in.
32)
Find the center of mass of a thin plate of constant density covering the given region.
33) The region bounded by y = x2 and y = 3
A) x = 0, y = 18
5
B) x = 0, y = 5
C) x = 0, y = 6
27
5
33)
D) x = 0, y = 9
5
34) The region enclosed by the parabolas y = - x2 + 8 and y = x2
16
C) x = 0, y = 5
B) x = 4, y = 0
A) x = 0, y = 8
34)
D) x = 0, y = 4
Find the centroid of the thin plate bounded by the graphs of the given functions. Use δ = 1 and M = area of the region
covered by the plate.
35) g(x) = x2 and f(x) = x + 2
35)
3
8
A) x = , y = 4
5
1
B) x = , y = 2
2
1
8
C) x = , y = 2
5
D) x = 2, y = 8
5
Solve the problem.
36) One end of a pool is a vertical wall 14 ft wide. What is the force exerted on this wall by the water if
it is 7 ft deep? The density of water is 62.4 lb/ft3 .
A) 3060 lb
B) 10,700 lb
C) 21,400 lb
D) 42,800 lb
37) A right triangular plate of base 8 m and height 4 m is submerged vertically, as shown below. Find
the force on one side of the plate if the top vertex is 1 m below the surface. (w = 9800 N/m 3 )
1 m
4 m
8 m
A) 810,000 N
B) 240,000 N
C) 420,000 N
7
36)
D) 410,000 N
37)
Answer Key
Testname: 2414‐2‐REVIEW
1) A
2) D
3) C
4) C
5) B
6) C
7) B
8) A
9) C
10) D
11) C
12) B
13) A
14) D
15) B
16) B
17) C
18) C
19) A
20) C
21) D
22) A
23) B
24) A
25) D
26) B
27) A
28) D
29) C
30) B
31) B
32) B
33) D
34) D
35) C
36) C
37) A
8