MATH 2414 - CALCULUS II TEST 2 REVIEW
Short Answer
1. Find the area of the region bounded by the
equations by integrating (i) with respect to x and
(ii) with respect to y.
6. The surface of a machine part is the region between
the graphs of y 1  x  and y 2  0.125x 2  k as
shown in the figure. Find k if the parabola is
tangent to the graph of y 1 . Round your answer to
three decimal places.
x  4  y2
x  y2
2. Find the area of the region bounded by the graphs
of the algebraic functions.
f  y   y 2  12 , g  y   0, y  12 ,
y  13
3. Find the area of the region bounded by the graphs
of the function f(x)  sin5x,


g x   cos 10x,
 x  . Round your answer
10
30
to three decimal places.
4. Find the area of the region bounded by the graphs
of the equations.
f(x)  sin x, g x   cos 2x,
7. Concrete sections for the new building have the
dimensions (in meters) and shape as shown in the
figure (the picture is not necessarily drawn to
scale). Find the area of the face of the section
superimposed on the rectangular coordinate system.
Round your answer to three decimal places.


x .
2
6
5. If the accumulation function F x  is given by
x
 1

F x     t 2  5  dt , evaluate F 9.
 11

0
1
Name: ________________________
ID: A
8. Set up and evaluate the integral that gives the
volume of the solid formed by revolving the region
x2
bounded by y  8 and y  16 
about the
16
x -axis.
15. Use the disk or shell method to find the volume of
the solid generated by revolving the region in the
first quadrant bounded by the graph of the equation
about the given line.
9. Find the volume of the solid generated by revolving
the region bounded by the graphs of the equations
y  10x 2 , y  0, and x  2 about the line y  40.
x
2
3
y
2
3
7
2
3
(i) the x-axis; (ii) the y-axis
10. Find the volume of the solid generated by revolving
the region bounded by the graphs of the equations
y  2x 2 , y  0 , and x  2 about the line x  2.
16. Find the arc length of the graph of the function
3
2 2
y  x  2 over the interval [14, 16].
3
11. Find the volume of the solid generated by revolving
the region bounded by the graphs of the equations
about the line y  8.
17. Find the arc length of the graph of the function
y
y  x, y  7, x  0
x3
1

over the interval [1,2].
6
2x
18. Find the arc length of the graph of the function
1
x   y  3  y over the interval 1  y  144.
3
1
x lying in the first
2
quadrant is revolved about the x -axis, a cone is
generated. Find the volume of the cone extending
from x  0 to x  26. Round your answer to two
decimal places.
12. If the portion of the line y 
19. Electrical wires suspended between two towers
form a catenary modeled by the equation
x
y  20cosh ,25  x  25 where x and y are
20
measured in meters. The towers are 50 meters
apart. Find the length of the suspended cable.
Round your answer to three decimal places.
13. Use the shell method to set up and evaluate the
integral that gives the volume of the solid generated
by revolving the plane region bounded by
2
1
y
e x /3 , y  0, x  0, and x  3 about the
3
y-axis. Round your answer to three decimal places.
14. Use the disk or the shell method to find the volume
of the solid generated by revolving the region
bounded by the graphs of the equations
17
y  2 , y  0, x  1, x  7 about the x-axis.
x
Round your answer to two decimal places.
2
Name: ________________________
ID: A
20. A barn is 75 feet long and 50 feet wide. A cross
section of the roof is the inverted catenary
 x
x 


 30

30

 . Find the number of
y  41  15  e  e







23. Neglecting air resistance and the weight of the
propellant, determine the work done in propelling a
12-ton satellite to a height of 100 miles above
Earth. Assume that the Earth has a radius of 4000
miles.
24. An open tank has the shape of a right circular cone.
The tank is 9 feet across the top and 8 feet high.
How much work is done in emptying the tank by
pumping the water over the top edge? Note: The
density of water is 62.4 lbs per cubic foot.
square feet of roofing on the barn. Round your
answer to the nearest integer.
25. Find the volume of the solid generated by rotating
2
the circle x 2   y  10   64 about the x-axis.
26. A circular plate of radius r feet is submerged
vertically in a tank of fluid that weighs w pounds
per cubic foot. The center of the circle is k k  r
feet below the surface of the fluid. The fluid force
on the surface of the plate is given by


F  wk  r 2  Find the fluid force on the circular


plate as shown in the figure given a  5 feet and
b  2 feet. Round your answer to one decimal
place.
21. Find the area of the surface generated by
revolving the curve about the x-axis.
y
1 3
x , 0  x  7.
7
22. Set up and evaluate the definite integral for the area
of the surface formed by revolving the graph of
y  9  x 2 about the y-axis. Round your answer to
three decimal places.
27. A porthole on a vertical side of a submarine
(submerged in seawater) is 2 square feet. Find the
fluid force on the porthole, assuming that the center
of the square is 14 feet below the surface.
3
ID: A
MATH 2414 - CALCULUS II
Answer Section
SHORT ANSWER
1. ANS:
125
A
6
PTS: 1
DIF: Medium
REF: 7.1.17b
OBJ: Calculate the area of a region bounded by two curves
NOT: Section 7.1
2. ANS:
4825
A
3
PTS: 1
DIF: Medium
REF: 7.1.33
OBJ: Calculate the area of a region bounded by several curves
NOT: Section 7.1
3. ANS:
0.260
PTS: 1
DIF: Medium
REF: 7.1.48
OBJ: Calculate the area of a region bounded by two curves
NOT: Section 7.1
4. ANS:
A
MSC: Application
MSC: Application
MSC: Application
33 / 2
4
PTS: 1
MSC: Application
5. ANS:
738
A
11
DIF: Medium
NOT: Section 7.1
REF: 7.1.48
PTS: 1
DIF: Easy
REF: 7.1.62
OBJ: Evaluate the accumulation function at a value
NOT: Section 7.1
6. ANS:
2.000
PTS: 1
DIF: Medium
REF: 7.1.96a
OBJ: Calculate slopes of tangent lines in applications
NOT: Section 7.1
1
OBJ: Calculate the area between two curves
MSC: Skill
MSC: Application
ID: A
7. ANS:
17.031 m2
PTS: 1
DIF: Medium
REF: 7.1.97a
OBJ: Calculate the area of a region bounded by several curves in applications
MSC: Application NOT: Section 7.1
8. ANS:
8 2 

2
 
2 


 
x
  64  dx  28672 2
  16 
V

 

16 
15
 


8 2 


PTS: 1
DIF: Medium
REF: 7.2.6
OBJ: Calculate the volume using the washer method of the solid formed by revolving a region about the x-axis
MSC: Application NOT: Section 7.2
9. ANS:
4,480

3
PTS: 1
DIF: Difficult
REF: 7.2.12c
OBJ: Calculate the volume using the washer method of the solid formed by revolving a region about a horizontal
line MSC:
Application
NOT: Section 7.2
10. ANS:
16

3
PTS: 1
DIF: Difficult
REF: 7.2.12d
OBJ: Calculate the volume using the disk method of the solid formed by revolving a region about a vertical line
MSC: Application NOT: Section 7.2
11. ANS:
490

3
PTS: 1
DIF: Medium
REF: 7.2.15
OBJ: Calculate the volume using the washer method of the solid formed by revolving a region about a horizontal
line MSC:
Application
NOT: Section 7.2
12. ANS:
4601.39
PTS: 1
DIF: Easy
REF: 7.2.57
OBJ: Calculate the volume using the disk method of the solid formed by revolving a region about the x-axis
MSC: Application NOT: Section 7.2
13. ANS:
2.917
PTS: 1
DIF: Medium
REF: 7.3.13
OBJ: Calculate the volume using the shell method of the solid formed by revolving a region about the y-axis
MSC: Application NOT: Section 7.3
2
ID: A
14. ANS:
301.76
PTS: 1
DIF: Medium
REF: 7.3.30b
OBJ: Calculate volumes of revolution by choosing an appropriate method
MSC: Application NOT: Section 7.3
15. ANS:
1,568
1,568
i 
 ; ii 

15
15
PTS:
OBJ:
MSC:
16. ANS:
2 
 17
3 
1
DIF: Medium
REF: 7.3.31a
Calculate volumes of revolution by choosing an appropriate method
Application NOT: Section 7.3

17  15 15 

PTS: 1
DIF: Medium
REF: 7.4.5
OBJ: Calculate the arc length of a curve over a given interval
NOT: Section 7.4
17. ANS:
17
12
PTS: 1
DIF: Medium
REF: 7.4.9
OBJ: Calculate the arc length of a curve over a given interval
NOT: Section 7.4
18. ANS:
1760
3
PTS: 1
DIF: Medium
REF: 7.4.16
OBJ: Calculate the arc length of a curve over a given interval
NOT: Section 7.4
19. ANS:
64.077 m
MSC: Application
MSC: Application
MSC: Application
PTS: 1
MSC: Application
20. ANS:
4200 square feet
DIF: Easy
NOT: Section 7.4
REF: 7.4.31
OBJ: Calculate arc lengths in applications
PTS: 1
MSC: Application
DIF: Medium
NOT: Section 7.4
REF: 7.4.32
OBJ: Calculate arc lengths in applications
3
ID: A
21. ANS:


3




2
7  442  1 







27
PTS: 1
DIF: Medium
REF: 7.4.37
OBJ: Calculate the area of a solid of revolution
NOT: Section 7.4
22. ANS:
117.319
PTS: 1
DIF: Medium
REF: 7.4.44
OBJ: Calculate the area of a solid of revolution
NOT: Section 7.4
23. ANS:
1,170.73 mi-ton
PTS: 1
DIF: Medium
REF: 7.5.15a
OBJ: Calculate work in problems involving propulsion
NOT: Section 7.5
24. ANS:
6,739.20 ft-lb
MSC: Application
MSC: Application
MSC: Application
PTS: 1
DIF: Difficult
REF: 7.5.21
OBJ: Calculate work in problems involving pumping liquids from containers
MSC: Application NOT: Section 7.5
25. ANS:
V  1,280 2
PTS: 1
DIF: Medium
REF: 7.6.52
OBJ: Calculate the volume of a solid of revolution using the Theorem of Pappus
MSC: Application NOT: Section 7.6
26. ANS:
5,489.0 lbs
PTS: 1
DIF: Easy
REF: 7.7.24a
OBJ: Calculate the fluid force on a submerged vertical surface
NOT: Section 7.7
27. ANS:
3,584lb
PTS: 1
DIF: Easy
REF: 7.7.27
OBJ: Calculate the fluid force on a submerged vertical surface
NOT: Section 7.7
4
MSC: Application
MSC: Application