Chapter 16
1.
Calculate the iterated integral.
1
1
0
0
  ( x  2 ye
2.
Compute
x
) dydx

16  x 2  y 2 dA , where D is the disk x 2  y 2  16 , by first identifying the integral as
D
the volume of a solid.
3.
Use polar coordinates to find the volume of the solid under the paraboloid z  x 2  y 2 and above the
disk x 2  y 2  49 .
Select the correct answer.
a. 1,277 .5
e. 1,267 .5
4.
b.
1,205 .5
c.
1,200 .5
d.
1,163 .5
Use polar coordinates to find the volume of the sphere of radius 6.
Select the correct answer.
824 .54
a.
924 .54
b.
c.
913 .26
d.
964 .54
e.
904 .78
5.
A cylindrical drill with radius 5 is used to bore a hole through the center of a sphere of radius 7. Find
the volume of the ring-shaped solid that remains.
6.
Evaluate
 x
2
y 2 dA
where D is the figure bounded by y  1, y  2, x  0, and x  y
D
7.
Find the center of mass of the lamina that occupies the region D and has the given density function, if
D is bounded by the parabola y  64  x 2 and the x-axis.
 ( x, y)  y
Select the correct answer.
a. (0, 36.57)
e. none of these
8.
b. (16, 56.57)
c.
(9, 40.57)
d.
(8, 46.57)
Find the area of the surface.
The part of the sphere x 2  y 2  z 2  9 that lies above the plane z = 2.
9.
Find the area of the part of the sphere x 2  y 2  z 2  4 z that lies inside the paraboloid z  x 2  y 2 .
10. Evaluate the integral.
3
4 1 z 2
  
1
1
1
5 ze3 y dxdzdy
11. Use polar coordinates to evaluate.

 

  y2
   y2
sin( x 2  y 2 ) dxdy
12. Find the moment of inertia about the y-axis for a cube of constant density 3 and side length 5 if one
vertex is located at the origin and three edges lie along the coordinate axes.
13. Find the region E for which the triple integral

(1  4 x 2  5 y 2  6 z 2 ) dV is a maximum.
E
14. Use cylindrical coordinates to evaluate the triple integral

y dV where E is the solid that lies
E
between the cylinders x 2  y 2  3 and x 2  y 2  7 above the xy-plane and below the
plane z  x  4 .
Select the correct answer.
a.
3.4
b.
9.19
c.
0
d.
15. Use spherical coordinates to evaluate the triple integral
8.57

xex
e.
2
 y2 z2
0.54
 dV
2
where E is the solid that
E
lies between the spheres x 2  y 2  z 2  9 and x 2  y 2  z 2  16 in the first octant.
2
2
2
16. Use spherical coordinates to find the volume of the solid that lies within the sphere x  y  z  9
above the xy-plane and below the cone z  x 2  y 2 .
17. Use spherical coordinates to find the moment of inertia of the solid homogeneous hemisphere of radius
3 and density 1 about a diameter of its base.
Select the correct answer.
a.
203.58
b.
198.08
c.
205.13
d.
213.5
e.
195.22
18. Use cylindrical or spherical coordinates, whichever seems more appropriate, to evaluate

z dV where E lies above the paraboloid z  x 2  y 2 and below the plane z  4 y .
E
Select the correct answer.
a.
160.28
b.
176.38
c.
167.55
d.
175.93
e. 175.37
19. Find the Jacobian of the transformation.
x
u
v
, y
2u  7v
4u  8v
2
2
2
20. Use spherical coordinates to find the volume above the cone z  x  y and inside the sphere
x 2  y 2  z 2  2az .
1
2
1.
e
2.
128

3
3.
c
4.
e
5.
492.5
6.
7/2
7.
a
8.
6
9.
4
10.
858818 .6
11.
2
12. 6250
13.
4x 2  5 y 2  6z 2  1
14. c
15.
e
16

256
16. 39.99
17. a
18. c
19. 0
20.
 a3
 e81

1.
Calculate the iterated integral.
1
1
0
0
  ( x  2 ye
2.
3.
x
) dydx
1
Find the volume of the “bumpy sphere” - the family of surfaces   1  sin  sin n with
5
and n = 3.
m=5
Evaluate the integral by making an appropriate change of variables.

e x  y dA , where R is given by the inequality | x |  | y |  3
R
4.
Calculate the double integral.
4  x2

5.
 ( x, y )
| 0  x  9, 0  y  1

Calculate the double integral.

R
6.
dA, R 
1 y2
R
 
 
x sin( x  y ) dA, R  0,
 0,

4
6 


Calculate the double integral.

xye y dA, R 
 ( x, y )
| 0  x  1, 0  y  1

R
7.
Find the volume of the solid in the first octant bounded by the cylinder z  9  y 2 and the plane x =
1.
8.
Use the given transformation to evaluate the integral.

( x  y ) dA ,
where R is the square with vertices (0, 0), (2, 3), (3, -2), (5, 1) and
R
x  2u  3v, y  3u  2v
9.
A lamina occupies the part of the disk x 2  y 2  25 in the first quadrant. Find its center of mass if the
density at any point is proportional to its distance from the x-axis.
10. Evaluate the iterated integral.
5
5
1
y
 
xy dxdy
11. Evaluate
 x
D
2
y 2 dA
where D is the figure bounded by y  1, y  2, x  0, and x  y .
12. Evaluate the double integral.

y 3 dA , where D is the triangular region with vertices (0, 1), (7, 0) and (1, 1).
D
13. Find the volume bounded by the cylinders x 2  y 2  25 and y 2  z 2  25 .
14. Evaluate the integral by reversing the order of integration.
1
4
 
0
2
e x dxdy
4y
15. Compute

4  x 2  y 2 dA ,
where D is the disk x 2  y 2  4, by first identifying the integral
D
as the volume of a solid.
16. Evaluate the integral by changing to polar coordinates.

e x
2
 y2
dA , where D is the region bounded by the semicircle x  4  y 2 and the y-axis.
D
2
2
17. Use polar coordinates to find the volume of the solid inside the cylinder x  y  9 and the ellipsoid
2x 2  2 y 2  z 2  36 .
Select the correct answer.
a. 260 .31
b. 301 .74
c. 261 .29
d. 292 .45
e. 284 .22
18. Find the Jacobian of the transformation.
x  6uv, y  4vw, z  5uw
19. Find the mass of the lamina that occupies the region D and has the given density function, if D is
bounded by the parabola x  y 2 and the line y  x  2 .
 ( x, y)  3
Select the correct answer.
a.
27
2
b.
9
2
c.
3
2
d.
7
2
e. none of those
20. Find the area of the surface. The part of the surface z  9  x 2  y 2 that lies above the xy-plane.
1.
e
1
2
2. 4.32
3. 60.11
4. 219.13
5. 0.11
6. 0.5
7. 18
8. 39
9.
 15 15 
 ,

16 
8
10. 72
11. 7 / 2
12.
1
5
13. 666.67
14.
e16  1
8
15.
16

3
1  e 
4
16.
2
17. d
18. 240 uvw
19. a
20. A(S ) 

6
(37 37  1)