1. Calculate the iterated integral.
2. Evaluate where is the figure bounded by and .
3. Evaluate the integral by changing to polar coordinates.
is the region bounded by the semicircle and the -axis.
4. Describe the r
...
1. Calculate the iterated integral.
2. Evaluate where is the figure bounded by and .
3. Evaluate the integral by changing to polar coordinates.
is the region bounded by the semicircle and the -axis.
4. Describe the region whose area is given by the integral.
5. Find the mass of the lamina that occupies the region and has the given density function. Round
your answer to two decimal places.
6. Find the area of the surface. The part of the surface that lies within the cylinder .
7. Use spherical coordinate to find the volume above the cone and inside sphere
.
8. Calculate the iterated integral.
9. Find the volume of the solid bounded in the first octanat bounded by the cylinder and
the planes .Stewart - Calculus ET 8e Chapter 15 Form A
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
10. Calculate the iterated integral.
11. Determine whether to use polar coordinates or rectangular coordinates to evaluate the integral
, where f is a continuous function. Then write an expression for the (iterated)
integral.
–6 –5 –4 –3 –2 –1 1 2 3 4 5 6 x
6 5 4 3 2 1
–1
–2
–3
–4
–5
–6
y
12. Evaluate the integral , where R is the annular region bounded by the circles
and by changing to polar coordinates.
13. Find the center of mass of the lamina of the region shown if the density of the circular lamina is
four times that of the rectangular lamina.
–2 2 x
1
–1
y
14. Find the mass and the center of mass of the lamina occupying the region R, where R is the region
bounded by the graphs of and and having the mass densityStewart - Calculus ET 8e Chapter 15 Form A
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
15. Find the mass and the moments of inertia and and the radii of gyration and for the
lamina occupying the region R, where R is the region bounded by the graphs of the equations
and and having the mass density
16. Find the area of the surface S where S is the part of the plane that lies above the
triangular region with vertices , and
17. Find the area of the surface S where S is the part of the sphere that lies inside the
cylinder
18. Sketch the solid bounded by the graphs of the equations and , and then
use a triple integral to find the volume of the solid.
19. Sketch the solid whose volume is given by the iterated integral
20. Find the center of mass of a homogeneous solid bounded by the paraboloid andStewart - Calculus ET 8e Chapter 15 Form A
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1.
2. 14
3.
4. half the region inside the loop of the four-leaved in quadrant I and IV
5.
6.
7.
8.
9. 18
10.
11. Polar,
12. 7
2
13.
14. ,
15. 64
5 , 16, 144 5 , 3
16.
17.Stewart - Calculus ET 8e Chapter 15 Form A
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
18.
19.
20.Stewart – Calculus ET 8e Chapter 15 Form B
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
1. Calculate the double integral. Round your answer to two decimal places.
2. Calculate the double integral. Round your answer to two decimal places.
3. Calculate the iterated integral.
4. Evaluate the integral by reversing the order of integration.
5. An agricultural sprinkler distributes water in a circular pattern of radius ft. It supplies water to
a depth of feet per hour at a distance of feet from the sprinkler. What is the total amount of
water supplied per hour to the region inside the circle of radius feet centered at the sprinkler?
6. Use polar coordinates to evaluate.
7. Evaluate the triple integral. Round your answer to one decimal place.
lies under the plane and above the region in the -plane bounded by the curves
, and .
8. Use spherical coordinate to find the volume above the cone and inside sphere
.Stewart – Calculus ET 8e Chapter 15 Form B
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
9. Evaluate the integral by making an appropriate change of variables. Round your answer to two
decimal places.
R is the parallelogram bounded by the lines
.
10. Find the volume of the solid bounded in the first octanat bounded by the cylinder and
the planes .
11. Evaluate the double integral , where
12. Evaluate the double integral , where is the region bounded by the graphs of
and .
13. Determine whether to use polar coordinates or rectangular coordinates to evaluate the integral
, where f is a continuous function. Then write an expression for the (iterated)
integral.
–6 –5 –4 –3 –2 –1 1 2 3 4 5 6 x
6 5 4 3 2 1
–1
–2
–3
–4
–5
–6
y
14. Find the center of mass of the system comprising masses mk located at the points Pk in a coordinate
plane. Assume that mass is measured in grams and distance is measured in centimeters.
m1 = 4, m2 = 3, m3 = 1
P1(3, –3), P2(5, –1), P3(2, –5)Stewart – Calculus ET 8e Chapter 15 Form B
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
15. Find the mass and the center of mass of the lamina occupying the region R, where R is the region
bounded by the graphs of and and having the mass density
16. Find the area of the surface S where S is the part of the surface that lies inside the cylinder
17. Find the area of the surface S where S is the part of the sphere that lies to the
right of the xz-plane and inside the cylinder
18. Sketch the solid bounded by the graphs of the equations and , and then
use a triple integral to find the volume of the solid.
19. Identify the surface with equation
20. Identify the surface with equationStewart – Calculus ET 8e Chapter 15 Form B
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1.
2.
3.
4.
5.
6.
7.
8.
9. 0.28
10. 18
11. 22
12. 189
20
13. Polar,
14.
15. ,
16.
17.Stewart – Calculus ET 8e Chapter 15 Form B
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
18.
19. Upper half of a right circular cone with vertex the origin and axis the positive z-axis
20. Sphere with radius 7 centered at the originStewart - Calculus ET 8e Chapter 15 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Select the correct answer for each question.
____ 1. Evaluate the double integral by first identifying it as the volume of a solid.
a.
b.
c.
d.
e.
____ 2. Use the Midpoint Rule with four squares of equal size to estimate the double integral.
a.
b.
c.
d.
e.
____ 3. Calculate the double integral.
a.
b.
c.
d.
e.
____ 4. Use polar coordinates to find the volume of the solid under the paraboloid and above
the disk .
a.
b.
c.
d.
e.Stewart - Calculus ET 8e Chapter 15 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 5. Evaluate the iterated integral by converting to polar coordinates. Round the answer to two decimal
places.
.
a.
b.
c.
d.
e.
____ 6. A swimming pool is circular with a -ft diameter. The depth is constant along east-west lines and
increases linearly from ft at the south end to ft at the north end. Find the volume of water in
the pool.
a.
b.
c.
d.
e.
____ 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.
a.
b.
c.
d.
e. None of theseStewart - Calculus ET 8e Chapter 15 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 8. Find the area of the surface.
The part of the surface that lies above the xy-plane.
a.
b. 2
75
c. 2
75
d. 2
75
e.
____ 9. Calculate the iterated integral.
a. 8
3
b. 4
c.
d.
e. None of theseStewart - Calculus ET 8e Chapter 15 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 10. Use a triple integral to find the volume of the solid bounded by and the planes and
.
a.
b.
c.
d.
e.
____ 11. Use cylindrical coordinates to evaluate where T is the solid bounded by the
cylinder and the planes and
a. 8
3
b. 4
c. 32
d. 48
____ 12. 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
____ 13. Use cylindrical or spherical coordinates, whichever seems more appropriate, to evaluate
where E lies above the paraboloid and below the plane .
a. 160.28
b. 176.38
c.
d. 175.93
e. 175.37Stewart - Calculus ET 8e Chapter 15 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 14. Use spherical coordinates.
Evaluate , where is the ball with center the origin and radius .
a.
b.
c.
d.
e. None of these
____ 15. Use spherical coordinates to evaluate where B is the ball
a. 10000
b. 10
c. 1000
d. 2000Stewart - Calculus ET 8e Chapter 15 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 16. The sketch of the solid is given below. Given , write the inequalities that describe it.
a.
b. None of these
c.
d.
e.
____ 17. Find the Jacobian of the transformation.
a.
b.
c.
d.
e.Stewart - Calculus ET 8e Chapter 15 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 18. Use the transformation to evaluate the integral
, where R is the region bounded by the ellipse .
a.
b.
c.
d.
e.
____ 19. Calculate the iterated integral.
a.
b.
c.
d.
e.Stewart - Calculus ET 8e Chapter 15 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 20. Evaluate the iterated integral.
a.
b.
c.
d.
e.Stewart - Calculus ET 8e Chapter 15 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1. E
2. C
3. E
4. E
5. A
6. E
7. A
8. C
9. A
10. A
11. A
12. B
13. C
14. A
15. A
16. D
17. C
18. E
19. E
20. AStewart - Calculus ET 8e Chapter 15 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Select the correct answer for each question.
____ 1. Evaluate the double integral by first identifying it as the volume of a solid.
a.
b.
c.
d.
e.
____ 2. Find the volume of the solid bounded by the surface and the planes
and coordinate planes.
a.
b.
c.
d.
e.
____ 3. Calculate the double integral.
a.
b.
c.
d.
e.
____ 4. Use polar coordinates to find the volume of the solid under the paraboloid and above
the disk .
a.
b.
c.
d.
e.Stewart - Calculus ET 8e Chapter 15 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 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
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 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
9Stewart - Calculus ET 8e Chapter 15 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 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. 24Stewart - Calculus ET 8e Chapter 15 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 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 theseStewart - Calculus ET 8e Chapter 15 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 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
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1. A
2. D
3. D
4. C
5. C
6. E
7. D
8. A
9. C
10. E
11. A
12. A
13. C
14. B
15. B
16. B
17. B
18. B
19. A
20. EStewart - Calculus ET 8e Chapter 15 Form E
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 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
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 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
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 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
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1. 3
13
2. D
3.
4.
5.
6. B
7. 2
9
8. B
9.
10. 8
3
11. A
12. 7
3
13. D
14. 0
15.
16. A
17. 625
4 k
18. B
19.
20.Stewart – Calculus ET 8e Chapter 15 Form F
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 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 densityStewart – Calculus ET 8e Chapter 15 Form F
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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.Stewart – Calculus ET 8e Chapter 15 Form F
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 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 ballStewart – Calculus ET 8e Chapter 15 Form F
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1. E
2.
3.
4.
5. C
6. ,
7.
8. A
9. 1
54
10.
11. 16
3
12. A
13. 19.63
14. B
15.
16. 1296
17.
18.
19. A
20. 14.1Stewart – Calculus ET 8e Chapter 15 Form G
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 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
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 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
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 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. 198.08Stewart – Calculus ET 8e Chapter 15 Form G
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1. B
2.
3.
4. 32
5. A
6.
7. D
8. 2
9
9.
10.
11. 8
3
12. A
13. E
14. 14
15. 0
16. B
17.
18.
19. 3.296
20. AStewart - Calculus ET 8e Chapter 15 Form H
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 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
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 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
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 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
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1. C
2.
3. E
4. D
5.
6. A
7.
8. ,
9. 27
2
10. B
11.
12. E
13. 7
3
14. 0
15.
16. E
17.
18. 3.296
19. 312
20.
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