Berkeley City College
Practice Problems
Math 3B - Calculus II - Chapter 8
Arc Length, Surface Area of Revolution, Hydrostatic Force, Centroids
Name___________________________________
Set up an integral for the length of the curve.
1) x = y1/3, 0 ≤ y ≤ 2
A)
∫
2
0
C)
∫
2
0
2) y =
A)
C)
9y4/3 + 1 dy
9y4/3
B)
3y2/3 + 1 dy
3y2/3
D)
∫
2
2
dy
9y4/3
2
1
dy
9y4/3
0
∫
0
1 - x8, - 1 ≤ x ≤ 1
4
4
∫
1/4
-1/4
1/4
∫
-1/4
5 - 4x8 dx
4(1 - x8)
B)
4 - 4x8 + 64x14 dx
4(1 - x8)
D)
∫
1/4
-1/4
1/4
∫
-1/4
4 + 64x14 dx
4
4 - 4x8 + 8x7 dx
4(1 - x8)
Find the area of the surface generated by revolving the curve about the indicated axis.
3
3) y = x 0 ≤ x ≤ 2; x-axis
9
A) 256 !
27
4) y =
B) 98 !
81
C) 98!
D) 1163 !
2187
C) 3!
D) 4!
3x - x2, 0.5 ≤ x ≤ 1.5; x-axis
A) 2!
B) !
5) x = 3 4 - y, 0 ≤ y ≤ 15/4; y-axis
A) 125 !
2
B) 125 - 5 10 !
2
C) 125 + 5 10 !
2
D) 5! 10
Instructor K. Pernell
1
Find the length of the curve.
6) x = 2 (y - 1) 3/2 between y = 4 to y = 9
3
A) 57
2
B) 31
3
C) 2
E) 38
3
D) 19
Set up an integral for the area of the surface generated by revolving the given curve about the
indicated axis.
7) xy = 3, 1 ≤ y ≤ 2; y-axis
A) 3!
∫
2 1
y
1 + 3y-4 dx
∫
2 1
y
1 + 9y-4 dx
1
C) 6!
1
B) 6!
∫
2 1
y
1 + 3y-4 dx
∫
2 1
y
1 + 9y-4 dx
1
D) 3!
1
Assume the region is part of a vertical side of a tank with water (δ = 62.4 pounds per cubic foot) at
the level shown. Find the total force exerted by the water against this region.
8)
y
y = 2x2
(1, 2)
x
A) 23.53 lb
B) 94.13 lb
C) 33.28 lb
2
D) 66.56 lb
9)
2 ft
4 ft
A) 1568.28 lb
B) 4704.85 lb
C) 784.14 lb
D) 3136.57 lb
Solve the problem.
10) A right triangular plate of base 8.0 m and height 4.0 m is submerged vertically in a fluid
with
density = 9800 N/m 3 . Find the force on one side of the plate if the top vertex is 1 m
below the surface.
1m
4.0 m
8.0 m
A) 410,000 N
B) 810,000 N
C) 240,000 N
D) 420,000 N
11) A rectangular sea aquarium observation window is 14.0 ft wide and 6.00 ft high. What is
the force on this window if the upper edge is 5.00 ft below the surface of the water. The
density of seawater is 64.0 lb/ft3.
A) 38,100 lb
B) 54,200 lb
C) 86,000 lb
D) 43,000 lb
12) An isosceles triangular plate is submerged vertically in seawater, with its base on the
bottom. The base is 6 ft long, and the height of the triangle is 6 ft. Find the force exerted
on one face of the plate if the water level is 2 ft above the base of the triangle. Seawater
weighs 64 lb/ft3. Round your answer to one decimal place if necessary.
A) 2048 lb
B) 1024 lb
C) 170.7 lb
3
D) 682.7 lb
Find the center of mass of a thin plate of constant density covering the given region.
13) The region bounded by y = x4, x = 2, and the x-axis
A) x = 5 , y = 40
3
9
B) x = 1 , y = 8
3
9
C) x = 5 , y = 20
6
9
D) x = 7 , y = 20
6
9
14) The region bounded by the parabola y = 9 - x2 and the x-axis
A) x = 0, y = 18
5
B) x = 0, y = 5
18
C) x = 0, y = 648
5
15) The region bounded by the x-axis and the semicircle y =
64 - x2
A) x = 0, y = 32
3!
B) x = 0, y = 8
3!
C) x = 32 , y = 32
3!
3!
D) x = 8 , y = 0
3!
16) The region bounded by the x-axis and the curve y = 5sin x, 0 ≤ x ≤ !
A) x = ! , y = 5!
2
2
B) x = !, y = 5!
4
C) x = ! , y = 5!
2
8
D) x = ! , y = 25!
2
4
17) The region bounded by y = 10 - x and the axes
A) x = 50, y = 50
B) x = 10 , y = 10
3
3
C) x = 10, y = 10
D) x = 500 , y = 500
3
3
4
D) x = 18 , y = 0
5
Solve the problem.
18) The lower edge of a dam is defined by the parabola (see figure). Use a coordinate system
with y = 0 at the bottom of the dam to determine the total force on the dam. Lengths are
measured in meters.
(12, 16)
2
y= x
9
A) 1,784,036 N
B) 32,112,640 N
5
C) 8,028,160 N
D) 16,056,320 N
Answer Key
Testname: MATH3B_PRACTICEPROBLEMS_APPSOFINTEGRALS_CH8
1) A
2) C
3) B
4) C
5) B
6) E
7) C
8) D
9) D
10) B
11) D
12) D
13) A
14) A
15) A
16) C
17) B
18) D
6