Stewart – Calculus ET 8e Chapter 15 Form D
____ 5. Use polar coordinates to find the volume of the solid bounded by the paraboloid
and the plane .
a.
b.
c.
d.
e.
____ 6. Use polar coordinates to find the volume of the solid inside the cylinder and the
ellipsoid .
a.
b.
c.
d.
e.
____ 7. Find the mass and the center of mass of the lamina occupying the region R, where R is the
triangular region with vertices and , and having the mass density
a. ,
b.
10
,
c.
10
,
d. ,
Stewart – Calculus ET 8e Chapter 15 Form D
____ 8. An electric charge is spread over a rectangular region Find the
total charge on R if the charge density at a point in R (measured in coulombs per square
meter) is
a.
84
coulombs
b.
222
coulombs
c.
22
coulombs
d.
56
coulombs
____ 9. 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 and the x-axis.
a.
b.
c.
d.
e. None of these
____ 10. Find the area of the part of hyperbolic paraboloid that lies between the cylinders
and .
a.
b.
c.
1
9
d.
1
9
e.
1
9
Stewart – Calculus ET 8e Chapter 15 Form D
____ 11. Calculate the iterated integral.
a.
8
3
b. 4
c.
d.
e. None of these
____ 12. Use a triple integral to find the volume of the solid bounded by and the planes and
.
a.
b.
c.
d.
e.
____ 13. Evaluate the integral where and
with respect to x, y, and z, in that order.
a. 28
b. 36
c. 4
d. 24
Stewart – Calculus ET 8e Chapter 15 Form D
____ 14. Use cylindrical coordinates to evaluate where T is the solid bounded by the
cylinder and the planes and
a.
4
b.
8
3
c.
24
d.
16
____ 15. Use cylindrical coordinates to evaluate the triple integral
where E is the solid that lies between the cylinders and above the xy-plane
and below the plane .
a. 8.57
b. 0
c. 3.4
d. 9.19
e. 0.54
____ 16. Use spherical coordinates.
Evaluate , where is the ball with center the origin and radius .
a.
b.
c.
d.
e. None of these
Stewart – Calculus ET 8e Chapter 15 Form D
____ 17. Use the given transformation to evaluate the integral.
, where R is the region in the first quadrant bounded by the lines and the
hyperbolas .
a. 9.447
b. 3.296
c. 8.841
d. 4.447
e. 5.088
____ 18. Evaluate the iterated integral.
a.
b.
c.
d.
e.
____ 19. Use a computer algebra system to find the moment of inertia of the lamina that occupies the
region D and has the density function , if .
a. 14.1
b. 34.138
c. 24.105
d. 28.5
e. None of these
____ 20. Calculate the iterated integral. Round your answer to two decimal places.
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 15 Form D
Answer Key
Stewart – Calculus ET 8e Chapter 15 Form E
1. Find the volume under and above the region bounded by and .
____ 2. Find the volume of the solid bounded by the surface and the planes
and coordinate planes.
Select the correct answer.
a.
b.
c.
d.
e.
3. Calculate the double integral.
4. Evaluate the iterated integral by converting to polar coordinates. Round the answer to two decimal
places.
.
5. Use polar coordinates to find the volume of the solid inside the cylinder and the
ellipsoid .
Stewart – Calculus ET 8e Chapter 15 Form E
____ 6. Find the mass and the center of mass of the lamina occupying the region R, where R is the
triangular region with vertices and , and having the mass density
Select the correct answer.
a.
25
,
b. ,
c.
25
,
d. ,
7. Find the area of the part of hyperbolic paraboloid that lies between the cylinders
and .
____ 8. Find the area of the part of the sphere that lies inside the paraboloid .
Select the correct answer.
a.
b.
c.
d.
e.
9. Find the area of the surface. The part of the sphere that lies above the plane .
10. Calculate the iterated integral.
Stewart – Calculus ET 8e Chapter 15 Form E
____ 11. Use a triple integral to find the volume of the solid bounded by and the planes and
.
Select the correct answer.
a.
b.
c.
d.
e.
12. Evaluate where and T is the region bounded by the paraboloid
and the plane
____ 13. Use cylindrical coordinates to evaluate
Select the correct answer.
a.
12
b.
112
c.
3
2
d.
14
14. Use cylindrical coordinates to evaluate the triple integral
where E is the solid that lies between the cylinders and above the xy-plane
and below the plane .
15. Use cylindrical or spherical coordinates, whichever seems more appropriate, to evaluate
where E lies above the paraboloid and below the plane .
Stewart – Calculus ET 8e Chapter 15 Form E
____ 16. Use spherical coordinates to evaluate where B is the ball
Select the correct answer.
a. 2401
b. 343
c. 686
d. 7
17. Find the mass of a solid hemisphere of radius 5 if the mass density at any point on the solid is
directly proportional to its distance from the base of the solid.
____ 18. Use the given transformation to evaluate the integral.
, where R is the region in the first quadrant bounded by the lines and the
hyperbolas .
Select the correct answer.
a. 9.447
b. 3.296
c. 8.841
d. 4.447
e. 5.088
19. Evaluate the iterated integral.
20. Calculate the iterated integral. Round your answer to two decimal places.
Stewart – Calculus ET 8e Chapter 15 Form E
Answer Key
Stewart – Calculus ET 8e Chapter 15 Form F
____ 1. Evaluate the double integral by first identifying it as the volume of a solid.
Select the correct answer.
a.
b.
c.
d.
e.
2. Use the Midpoint Rule with four squares of equal size to estimate the double integral.
3. Estimate the volume of the solid that lies above the square and below the
elliptic paraboloid .
Divide into four equal squares and use the Midpoint rule.
4. Find the volume of the solid bounded by the surface and the planes
and coordinate planes.
____ 5. Use polar coordinates to find the volume of the solid bounded by the paraboloid
and the plane .
Select the correct answer.
a.
b.
c.
d.
e.
6. Find the mass and the center of mass of the lamina occupying the region R, where R is the
triangular region with vertices and , and having the mass density
Stewart – Calculus ET 8e Chapter 15 Form F
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 and the x-axis.
____ 8. Find the mass of the lamina that occupies the region D and has the given density function, if D is
bounded by the parabola and the line .
Select the correct answer.
a.
45
2
b.
5
2
c. 45
d.
3
e. None of these
9. Find the area of the surface.
The part of the surface that lies above the xy-plane.
10. Find the area of the surface. The part of the sphere that lies above the plane .
11. Calculate the iterated integral.