Stewart – Calculus ET 8e Chapter 15 Form F
____ 12. 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.
13. Find the mass of the solid S bounded by the paraboloid and the plane if S has
constant density 3.
____ 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 .
Select the correct answer.
a. 8.57
b. 0
c. 3.4
d. 9.19
e. 0.54
15. Use cylindrical coordinates to evaluate
where E is the region that lies inside the cylinder and between the planes
.
Round the answer to two decimal places.
16. Use spherical coordinates to evaluate where B is the ball
Stewart – Calculus ET 8e Chapter 15 Form F
17. The sketch of the solid is given below. Given , write the inequalities that describe it.
18. Calculate the iterated integral.
____ 19. Evaluate the iterated integral.
Select the correct answer.
a.
b.
c.
d.
e.
20. 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 .
Stewart – Calculus ET 8e Chapter 15 Form F
Answer Key
Stewart – Calculus ET 8e Chapter 15 Form G
____ 1. Calculate the double integral.
Select the correct answer.
a.
b.
c.
d.
e.
2. Use polar coordinates to find the volume of the solid under the paraboloid and above
the disk .
3. Use polar coordinates to find the volume of the solid bounded by the paraboloid
and the plane .
4. Use a double integral to find the area of the region R where R is bounded by the circle
____ 5. 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
Select the correct answer.
a.
804
coulombs
b.
91
coulombs
c.
300
coulombs
d.
265
coulombs
6. 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.
Stewart – Calculus ET 8e Chapter 15 Form G
____ 7. 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.
3
2
b.
2
c. 27
d.
27
2
e. None of these
8. Find the area of the part of hyperbolic paraboloid that lies between the cylinders
and .
9. Find the area of the surface. Round your answer to three decimal places.
10. Find the area of the part of the sphere that lies inside the paraboloid .
11. Calculate the iterated integral.
Stewart – Calculus ET 8e Chapter 15 Form G
____ 12. Evaluate the integral where and
with respect to x, y, and z, in that order.
Select the correct answer.
a. 4
b. 28
c. 36
d. 24
____ 13. Find the mass of the solid S bounded by the paraboloid and the plane if S has
constant density 3. Select the correct answer.
a. 16.25
b. 15.07
c. 24.91
d. 13.92
e. 19.63
14. Use cylindrical coordinates to evaluate
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 .
____ 16. Use spherical coordinates to find the moment of inertia of the solid homogeneous hemisphere of
radius and density 1 about a diameter of its base. Select the correct answer.
a. 205.13
b.
c. 195.22
d. 213.5
e. 198.08
Stewart – Calculus ET 8e Chapter 15 Form G
17. Find the Jacobian of the transformation.
18. Use the transformation to evaluate the integral
, where R is the region bounded by the ellipse .
19. Use the given transformation to evaluate the integral.
, where R is the region in the first quadrant bounded by the lines and the
hyperbolas .
____ 20. Evaluate the iterated integral. Select the correct answer.
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 15 Form G
Answer Key
Stewart – Calculus ET 8e Chapter 15 Form H
____ 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. 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.
____ 3. Use polar coordinates to find the volume of the solid under the paraboloid and above
the disk . Select the correct answer.
a.
b.
c.
d.
e.
____ 4. Use a double integral to find the area of the region R where R is bounded by the circle
Select the correct answer.
a.
64
b.
16
c.
8
d.
32
5. Use polar coordinates to find the volume of the sphere of radius . Round to two decimal places.
Stewart – Calculus ET 8e Chapter 15 Form H
____ 6. For which of the following regions would you use rectangular coordinates?
a.
c.
b.
7. Find the center of mass of a lamina in the shape of an isosceles right triangle with equal sides of
length if the density at any point is proportional to the square of the distance from the
vertex opposite the hypotenuse. Assume the vertex opposite the hypotenuse is located at , and
that the sides are along the positive axes.
8. 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
9. 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 .
____ 10. Find the area of the part of the sphere that lies inside the paraboloid .
Select the correct answer.
a.
b.
c.
Stewart – Calculus ET 8e Chapter 15 Form H
d.
e.
11. Find the area of the surface. The part of the sphere that lies above the plane .
____ 12. Find the mass of the solid S bounded by the paraboloid and the plane if S has
constant density 3. Select the correct answer.
a. 16.25
b. 15.07
c. 24.91
d. 13.92
e. 19.63
13. Evaluate where and T is the region bounded by the paraboloid
and the plane
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 coordinates to evaluate
where E is the region that lies inside the cylinder and between the planes
.
Round the answer to two decimal places.
Stewart – Calculus ET 8e Chapter 15 Form H
____ 16. The sketch of the solid is given below. Given , write the inequalities that describe it.
Select the correct answer.
a. None of these
b.
c.
d.
e.
17. Find the Jacobian of the transformation.
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 .
19. Use the given transformation to evaluate the integral.
, where R is the square with vertices (0, 0), (4, 6), (6, ), (10, 2) and
20. Calculate the iterated integral.
Stewart – Calculus ET 8e Chapter 15 Form H
Answer Key