MATH 2414 - CALCULUS II TEST 2 REVIEW

MATH 2414 - CALCULUS II TEST 2 REVIEW

Short Answer

1. Find the area of the region bounded by the 6. The surface of a machine part is the region between
equations by integrating (i) with respect to x and the graphs of y1  x and y2  0.125x2  k as
(ii) with respect to y. shown in the figure. Find k if the parabola is
tangent to the graph of y1 . Round your answer to
x  4y2 three decimal places.
x y2
7. Concrete sections for the new building have the
2. Find the area of the region bounded by the graphs dimensions (in meters) and shape as shown in the
of the algebraic functions. figure (the picture is not necessarily drawn to
scale). Find the area of the face of the section
f  y   y2  12 , g y   0, y  12 , y  13 superimposed on the rectangular coordinate system.
Round your answer to three decimal places.
3. Find the area of the region bounded by the graphs

of the function f(x)  sin 5x,

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,  x  .
2 6

5. If the accumulation function Fx is given by

F x   x  1 t2  5 dt , evaluate F9.
11


0

1

Name: ________________________ ID: A

8. Set up and evaluate the integral that gives the 15. Use the disk or shell method to find the volume of
the solid generated by revolving the region in the
volume of the solid formed by revolving the region first quadrant bounded by the graph of the equation
about the given line.
bounded by y 8 and y  16  x2 about the
16

x -axis. 22 2

9. Find the volume of the solid generated by revolving x3 y3 73
the region bounded by the graphs of the equations
y  10x2 , y  0, and x  2 about the line y  40. (i) the x-axis; (ii) the y-axis

10. Find the volume of the solid generated by revolving 16. Find the arc length of the graph of the function
the region bounded by the graphs of the equations
y  2x2 , y  0, and x  2 about the line x  2. 3

y  2 x2 2 over the interval [14, 16].
3

11. Find the volume of the solid generated by revolving 17. Find the arc length of the graph of the function
the region bounded by the graphs of the equations
about the line y  8. y  x3 1 over the interval [1,2].
6 2x

y  x, y  7, x  0 18. Find the arc length of the graph of the function

12. If the portion of the line y  1 x lying in the first x  1  y  3 y over the interval 1  y  144.
2 3

quadrant is revolved about the x-axis, a cone is 19. Electrical wires suspended between two towers

generated. Find the volume of the cone extending form a catenary modeled by the equation

from x  0 to x  26. Round your answer to two x
20
decimal places. y  20 cosh , 25  x  25 where x and y are

13. Use the shell method to set up and evaluate the measured in meters. The towers are 50 meters
integral that gives the volume of the solid generated
by revolving the plane region bounded by apart. Find the length of the suspended cable.

y  1 ex2 /3 , y  0, x  0, and x  3 about the Round your answer to three decimal places.
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

y 17 , y  0, x  1, x  7 about the x-axis.
x2

Round your answer to two decimal places.

2

Name: ________________________ ID: A

20. A barn is 75 feet long and 50 feet wide. A cross 23. Neglecting air resistance and the weight of the
propellant, determine the work done in propelling a
section of the roof is the inverted catenary 12-ton satellite to a height of 100 miles above
15 e  . Earth. Assume that the Earth has a radius of 4000
x  x miles.
30 30
y  41   e Find the number of 24. An open tank has the shape of a right circular cone.
The tank is 9 feet across the top and 8 feet high.
square feet of roofing on the barn. Round your How much work is done in emptying the tank by
pumping the water over the top edge? Note: The
answer to the nearest integer. density of water is 62.4 lbs per cubic foot.

21. Find the area of the surface generated by 25. Find the volume of the solid generated by rotating
revolving the curve about the x-axis. the circle x2   y  10  2  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 r2  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.

y  1 x3, 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  x2 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:

A  125
6

PTS: 1 DIF: Medium REF: 7.1.17b

OBJ: Calculate the area of a region bounded by two curves MSC: Application

NOT: Section 7.1

2. ANS:

A  4825
3

PTS: 1 DIF: Medium REF: 7.1.33

OBJ: Calculate the area of a region bounded by several curves MSC: Application

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 MSC: Application

NOT: Section 7.1

4. ANS:

A  33 / 2
4

PTS: 1 DIF: Medium REF: 7.1.48 OBJ: Calculate the area between two curves
NOT: Section 7.1
MSC: Application

5. ANS:

A  738
11

PTS: 1 DIF: Easy REF: 7.1.62

OBJ: Evaluate the accumulation function at a value MSC: Skill

NOT: Section 7.1

6. ANS:

2.000

PTS: 1 DIF: Medium REF: 7.1.96a

OBJ: Calculate slopes of tangent lines in applications MSC: Application

NOT: Section 7.1

1

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:

V 8 2  16  x2  2  64 dx  28672 2
2 16 15


8

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:

i 1, 568  ; ii 1, 568 
15 15

PTS: 1 DIF: Medium REF: 7.3.31a

OBJ: Calculate volumes of revolution by choosing an appropriate method

MSC: Application NOT: Section 7.3

16. ANS:

2  17 17  15 15 
3

PTS: 1 DIF: Medium REF: 7.4.5

OBJ: Calculate the arc length of a curve over a given interval MSC: Application

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 MSC: Application

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 MSC: Application

NOT: Section 7.4

19. ANS:

64.077 m

PTS: 1 DIF: Easy REF: 7.4.31 OBJ: Calculate arc lengths in applications
MSC: Application NOT: Section 7.4

20. ANS:
4200 square feet

PTS: 1 DIF: Medium REF: 7.4.32 OBJ: Calculate arc lengths in applications

MSC: Application NOT: Section 7.4

3

ID: A

21. ANS:

7 442 3  1
2

27 

PTS: 1 DIF: Medium REF: 7.4.37

OBJ: Calculate the area of a solid of revolution MSC: Application

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 MSC: Application

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 MSC: Application

NOT: Section 7.5

24. ANS:

6,739.20 ft-lb

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 MSC: Application

NOT: Section 7.7

27. ANS:

3, 584 lb

PTS: 1 DIF: Easy REF: 7.7.27

OBJ: Calculate the fluid force on a submerged vertical surface MSC: Application

NOT: Section 7.7

4