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Deck 15: Multiple Integrals
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Question
Evaluate 
where 
and T is the region bounded by the paraboloid 
and the plane 
A)
B)
C)
D)
where and T is the region bounded by the paraboloid and the plane A)
B)
C)
D)
Question
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 
.
R is the parallelogram bounded by the lines . Question
Use the given transformation to evaluate the integral. 
, where R is the square with vertices (0, 0), (4, 6), (6, 
), (10, 2) and 
A)42
B)208
C)343
D)312
E)52
, where R is the square with vertices (0, 0), (4, 6), (6, ), (10, 2) and A)42
B)208
C)343
D)312
E)52
Question
The sketch of the solid is given below. Given 
, write the inequalities that describe it. 
A)
B)
C)
D)None of these
E)
, write the inequalities that describe it. A)
B)
C)
D)None of these
E)
Question
Use spherical coordinates. Evaluate 
, where 
is the ball with center the origin and radius 
.
A)
B)
C)
D)
E)None of these
, where is the ball with center the origin and radius .A)
B)
C)
D)
E)None of these
Question
Use spherical coordinates to find the volume of the solid that lies within the sphere 
above the xy-plane and below the cone 
. Round the answer to two decimal places.
above the xy-plane and below the cone . Round the answer to two decimal places. Question
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.
A)
k 
B)
k 
C)
k 
D)
k 
A)
k B)
k C)
k D)
k Question
Use cylindrical coordinates to evaluate 
where T is the solid bounded by the cylinder 
and the planes 
and 
A)
B)
C)
D)
where T is the solid bounded by the cylinder and the planes and A)
B)
C)
D)
Question
Use the transformation 
to evaluate the integral 
, where R is the region bounded by the ellipse 
.
A)
B)
C)
D)
E)
to evaluate the integral , where R is the region bounded by the ellipse .A)
B)
C)
D)
E)
Question
Identify the surface with equation 
Question
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)8.841
B)3.296
C)4.447
D)5.088
E)9.447
, where R is the region in the first quadrant bounded by the lines and the hyperbolas .A)8.841
B)3.296
C)4.447
D)5.088
E)9.447
Question
Use cylindrical coordinates to evaluate the triple integral 
where E is the solid that lies between the sphere 
and 
in the first octant.
where E is the solid that lies between the sphere and in the first octant. Question
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.
A)195.22
B)205.13
C)198.08
D)213.5
E)
and density 1 about a diameter of its base.A)195.22
B)205.13
C)198.08
D)213.5
E)
Question
Use spherical coordinates to evaluate 
where B is the ball 
A)512
B)64
C)1024
D)8
where B is the ball A)512
B)64
C)1024
D)8
Question
Find the Jacobian of the transformation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Find the Jacobian of the transformation. 
Question
Find the moment of inertia with respect to a diameter of the base of a solid hemisphere of radius 3 with constant mass density function 
Question
Identify the surface with equation 
Question
Use spherical coordinate to find the volume above the cone 
and inside sphere 
.
and inside sphere . Question
Identify the surface with equation 
Question
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 
.
lies under the plane and above the region in the -plane bounded by the curves , and . Question
Use cylindrical coordinates to find the volume of the solid that the cylinder 
cuts out of the sphere of radius 3 centered at the origin.
cuts out of the sphere of radius 3 centered at the origin. Question
Evaluate the integral 
where 
and 
with respect to x, y, and z, in that order.
A)120
B)500
C)620
D)180
where and with respect to x, y, and z, in that order.A)120
B)500
C)620
D)180
Question
Find the region E for which the triple integral 
is a maximum.
is a maximum. Question
Evaluate the iterated integral 
Question
Use cylindrical coordinates to evaluate 
A)
B)
C)
D)
A)
B)
C)
D)
Question
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)3.4
B)0
C)8.57
D)0.54
E)9.19
where E is the solid that lies between the cylinders and above the xy-plane and below the plane .A)3.4
B)0
C)8.57
D)0.54
E)9.19
Question
Calculate the iterated integral. 
A)
B)8
C)
D)
E)None of these
A)
B)8
C)
D)
E)None of these
Question
Use a triple integral to find the volume of the solid bounded by 
and the planes 
and 
.
A)
B)
C)
D)
E)
and the planes and .A)
B)
C)
D)
E)
Question
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.
A)
B)2218.41
C)2931.90
D)2431.90
E)2818.41
where E is the region that lies inside the cylinder and between the planes . Round the answer to two decimal places.A)
B)2218.41
C)2931.90
D)2431.90
E)2818.41
Question
The joint density function for random variables 
and 
is 
for 
and 
otherwise. Find the value of the constant 
.
Round the answer to the nearest thousandth.
and is for and otherwise. Find the value of the constant .Round the answer to the nearest thousandth.
Question
Find the mass of the solid E, if E is the cube given by 
and the density function 
is 
.
and the density function is . Question
Find the center of mass of a homogeneous solid bounded by the paraboloid 
and 
and Question
The joint density function for a pair of random variables 
and 
is given. 
Find the value of the constant 
.
and is given. Find the value of the constant . Question
Express the integral as an iterated integral of the form 
where E is the solid bounded by the surfaces 
where E is the solid bounded by the surfaces Question
Use cylindrical or spherical coordinates, whichever seems more appropriate, to evaluate 
where E lies above the paraboloid 
and below the plane 
.
A)
B)
160.28
C)175.37
D)176.38
E)175.93
where E lies above the paraboloid and below the plane .A)
B)
160.28C)175.37
D)176.38
E)175.93
Question
Find the mass of the solid S bounded by the paraboloid 
and the plane 
if S has constant density 3.
A)13.92
B)15.07
C)19.63
D)16.25
E)24.91
and the plane if S has constant density 3.A)13.92
B)15.07
C)19.63
D)16.25
E)24.91
Question
Express the volume of the wedge in the first octant that is cut from the cylinder 
by the planes 
and 
as an iterated integral with respect to 
, then to 
, then to 
.
by the planes and as an iterated integral with respect to , then to , then to . Question
Find the moment of inertia about the y-axis for a cube of constant density 3 and side length 
if one vertex is located at the origin and three edges lie along the coordinate axes.
if one vertex is located at the origin and three edges lie along the coordinate axes. Question
Evaluate the triple integral. Round your answer to one decimal place. 
Question
Set up, but do not evaluate, the iterated integral giving the mass of the solid T bounded by the cylinder 
in the first octant and the plane 
having mass density given by 
in the first octant and the plane having mass density given by Question
Find the area of the surface S where S is the part of the surface 
that lies inside the cylinder 
that lies inside the cylinder Question
Sketch the solid whose volume is given by the integral 
Evaluate the integral.
Evaluate the integral. Question
Describe the region whose area is given by the integral. 
Question
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 
that lies to the right of the xz-plane and inside the cylinder Question
Find the area of the part of the sphere 
that lies inside the paraboloid 
.
A)
B)
C)
D)
E)
that lies inside the paraboloid .A)
B)
C)
D)
E)
Question
Find the area of the surface. The part of the sphere 
that lies above the plane 
.
A)
B)
C)
D)
E)
that lies above the plane .A)
B)
C)
D)
E)
Question
Find the area of the surface. The part of the surface 
that lies within the cylinder 
.
that lies within the cylinder . Question
Find the area of the part of hyperbolic paraboloid 
that lies between the cylinders 
and 
.
A)
B)
C)
D)
E)
that lies between the cylinders and .A)
B)
C)
D)
E)
Question
Find the area of the surface. The part of the surface 
that lies above the xy-plane.
A)
B)
C)
D)
E)
that lies above the xy-plane.A)
B)
C)
D)
E)
Question
Express the triple integral 
as an iterated integral in six different ways using different orders of integration where T is the solid bounded by the planes 
and 
as an iterated integral in six different ways using different orders of integration where T is the solid bounded by the planes and Question
Sketch the solid whose volume is given by the iterated integral 
Question
Find the area of the surface. Round your answer to three decimal places. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Find the area of the part of the plane 
that lies in the first octant.
that lies in the first octant. Question
Find the area of the surface S where S is the part of the sphere 
that lies inside the cylinder 
that lies inside the cylinder Question
Sketch the solid bounded by the graphs of the equations 
and 
, and then use a triple integral to find the volume of the solid.
and , and then use a triple integral to find the volume of the solid. Question
Find the area of the part of the plane 
that lies inside the cylinder 
.
that lies inside the cylinder . Question
Find the area of the surface S where S is the part of the plane 
that lies above the triangular region with vertices 
, and 
that lies above the triangular region with vertices , and Question
Find the exact area of the surface. 
.
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
Question
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 these
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 these
Question
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
and the x-axis. A)
B)
C)
D)
E)None of these
Question
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 
and and having the mass density Question
Question
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)
, 
C)
, 
D)
, 
and , and having the mass density A)
, B)
, C)
, D)
, Question
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 
. 
A)
B)27
C)
D)
E)None of these
and the line . A)
B)27
C)
D)
E)None of these
Question
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 the equations 
and 
and having the mass density 
and and having the mass density Question
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 rectangular region with vertices 
and 
, and having uniform density 
and and the radii of gyration and for the lamina occupying the region R, where R is the rectangular region with vertices and , and having uniform density Question
Use polar coordinates to find the volume of the solid inside the cylinder 
and the ellipsoid 
.
A)
B)
C)
D)
E)
and the ellipsoid .A)
B)
C)
D)
E)
Question
A lamina occupies the part of the disk 
in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis.
in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis. Question
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. 
Question
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 
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 Question
Use polar coordinates to find the volume of the solid under the paraboloid 
and above the disk 
.
A)
B)
C)
D)
E)
and above the disk .A)
B)
C)
D)
E)
Question
Find the mass of the lamina that occupies the region 
and has the given density function. Round your answer to two decimal places. 
and has the given density function. Round your answer to two decimal places. Question
Use polar coordinates to find the volume of the sphere of radius 
. Round to two decimal places.
A)
B)
C)
D)
E)
. Round to two decimal places.A)
B)
C)
D)
E)
Question
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)
-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)
Question
Use polar coordinates to find the volume of the solid bounded by the paraboloid 
and the plane 
.
A)
B)
C)
D)
E)
and the plane .A)
B)
C)
D)
E)
Question
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)
coulombs
B)
coulombs
C)
coulombs
D)
coulombs
Find the total charge on R if the charge density at a point in R (measured in coulombs per square meter) is A)
coulombsB)
coulombsC)
coulombsD)
coulombs Question
Use a double integral to find the area of the region R where R is bounded by the circle 
A)
B)
C)
D)
A)
B)
C)
D)
Question
Evaluate the integral by changing to polar coordinates. 
is the region bounded by the semicircle 
and the 
-axis.
is the region bounded by the semicircle and the -axis. Question
Evaluate the iterated integral by converting to polar coordinates. Round the answer to two decimal places. 
.
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
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Deck 15: Multiple Integrals
1
Evaluate where and T is the region bounded by the paraboloid and the plane
A)
B)
C)
D)
where and T is the region bounded by the paraboloid and the plane A)
B)
C)
D)
2
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 .
R is the parallelogram bounded by the lines .
3
Use the given transformation to evaluate the integral. , where R is the square with vertices (0, 0), (4, 6), (6, ), (10, 2) and
A)42
B)208
C)343
D)312
E)52
, where R is the square with vertices (0, 0), (4, 6), (6, ), (10, 2) and A)42
B)208
C)343
D)312
E)52
312
4
The sketch of the solid is given below. Given , write the inequalities that describe it.
A)
B)
C)
D)None of these
E)
, write the inequalities that describe it. A)
B)
C)
D)None of these
E)
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5
Use spherical coordinates. Evaluate , where is the ball with center the origin and radius .
A)
B)
C)
D)
E)None of these
, where is the ball with center the origin and radius .A)
B)
C)
D)
E)None of these
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6
Use spherical coordinates to find the volume of the solid that lies within the sphere above the xy-plane and below the cone . Round the answer to two decimal places.
above the xy-plane and below the cone . Round the answer to two decimal places. Unlock Deck
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7
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.
A) k
B) k
C) k
D) k
A)
k B)
k C)
k D)
k Unlock Deck
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8
Use cylindrical coordinates to evaluate where T is the solid bounded by the cylinder and the planes and
A)
B)
C)
D)
where T is the solid bounded by the cylinder and the planes and A)
B)
C)
D)
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9
Use the transformation to evaluate the integral , where R is the region bounded by the ellipse .
A)
B)
C)
D)
E)
to evaluate the integral , where R is the region bounded by the ellipse .A)
B)
C)
D)
E)
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10
Identify the surface with equation
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11
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)8.841
B)3.296
C)4.447
D)5.088
E)9.447
, where R is the region in the first quadrant bounded by the lines and the hyperbolas .A)8.841
B)3.296
C)4.447
D)5.088
E)9.447
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12
Use cylindrical coordinates to evaluate the triple integral where E is the solid that lies between the sphere and in the first octant.
where E is the solid that lies between the sphere and in the first octant. Unlock Deck
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13
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.
A)195.22
B)205.13
C)198.08
D)213.5
E)
and density 1 about a diameter of its base.A)195.22
B)205.13
C)198.08
D)213.5
E)
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14
Use spherical coordinates to evaluate where B is the ball
A)512
B)64
C)1024
D)8
where B is the ball A)512
B)64
C)1024
D)8
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15
Find the Jacobian of the transformation.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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16
Find the Jacobian of the transformation.
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17
Find the moment of inertia with respect to a diameter of the base of a solid hemisphere of radius 3 with constant mass density function
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18
Identify the surface with equation
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19
Use spherical coordinate to find the volume above the cone and inside sphere .
and inside sphere . Unlock Deck
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20
Identify the surface with equation
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21
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 .
lies under the plane and above the region in the -plane bounded by the curves , and . Unlock Deck
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22
Use cylindrical coordinates to find the volume of the solid that the cylinder cuts out of the sphere of radius 3 centered at the origin.
cuts out of the sphere of radius 3 centered at the origin. Unlock Deck
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23
Evaluate the integral where and with respect to x, y, and z, in that order.
A)120
B)500
C)620
D)180
where and with respect to x, y, and z, in that order.A)120
B)500
C)620
D)180
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24
Find the region E for which the triple integral is a maximum.
is a maximum. Unlock Deck
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25
Evaluate the iterated integral
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26
Use cylindrical coordinates to evaluate
A)
B)
C)
D)
A)
B)
C)
D)
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27
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)3.4
B)0
C)8.57
D)0.54
E)9.19
where E is the solid that lies between the cylinders and above the xy-plane and below the plane .A)3.4
B)0
C)8.57
D)0.54
E)9.19
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28
Calculate the iterated integral.
A)
B)8
C)
D)
E)None of these
A)
B)8
C)
D)
E)None of these
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29
Use a triple integral to find the volume of the solid bounded by and the planes and .
A)
B)
C)
D)
E)
and the planes and .A)
B)
C)
D)
E)
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30
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.
A)
B)2218.41
C)2931.90
D)2431.90
E)2818.41
where E is the region that lies inside the cylinder and between the planes . Round the answer to two decimal places.A)
B)2218.41
C)2931.90
D)2431.90
E)2818.41
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31
The joint density function for random variables and is for and otherwise. Find the value of the constant .
Round the answer to the nearest thousandth.
and is for and otherwise. Find the value of the constant .Round the answer to the nearest thousandth.
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32
Find the mass of the solid E, if E is the cube given by and the density function is .
and the density function is . Unlock Deck
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33
Find the center of mass of a homogeneous solid bounded by the paraboloid and
and Unlock Deck
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34
The joint density function for a pair of random variables and is given. Find the value of the constant .
and is given. Find the value of the constant . Unlock Deck
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35
Express the integral as an iterated integral of the form where E is the solid bounded by the surfaces
where E is the solid bounded by the surfaces Unlock Deck
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36
Use cylindrical or spherical coordinates, whichever seems more appropriate, to evaluate where E lies above the paraboloid and below the plane .
A)
B) 160.28
C)175.37
D)176.38
E)175.93
where E lies above the paraboloid and below the plane .A)
B)
160.28C)175.37
D)176.38
E)175.93
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37
Find the mass of the solid S bounded by the paraboloid and the plane if S has constant density 3.
A)13.92
B)15.07
C)19.63
D)16.25
E)24.91
and the plane if S has constant density 3.A)13.92
B)15.07
C)19.63
D)16.25
E)24.91
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38
Express the volume of the wedge in the first octant that is cut from the cylinder by the planes and as an iterated integral with respect to , then to , then to .
by the planes and as an iterated integral with respect to , then to , then to . Unlock Deck
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39
Find the moment of inertia about the y-axis for a cube of constant density 3 and side length if one vertex is located at the origin and three edges lie along the coordinate axes.
if one vertex is located at the origin and three edges lie along the coordinate axes. Unlock Deck
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40
Evaluate the triple integral. Round your answer to one decimal place.
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41
Set up, but do not evaluate, the iterated integral giving the mass of the solid T bounded by the cylinder in the first octant and the plane having mass density given by
in the first octant and the plane having mass density given by Unlock Deck
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42
Find the area of the surface S where S is the part of the surface that lies inside the cylinder
that lies inside the cylinder Unlock Deck
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43
Sketch the solid whose volume is given by the integral Evaluate the integral.
Evaluate the integral. Unlock Deck
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44
Describe the region whose area is given by the integral.
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45
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
that lies to the right of the xz-plane and inside the cylinder Unlock Deck
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46
Find the area of the part of the sphere that lies inside the paraboloid .
A)
B)
C)
D)
E)
that lies inside the paraboloid .A)
B)
C)
D)
E)
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47
Find the area of the surface. The part of the sphere that lies above the plane .
A)
B)
C)
D)
E)
that lies above the plane .A)
B)
C)
D)
E)
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48
Find the area of the surface. The part of the surface that lies within the cylinder .
that lies within the cylinder . Unlock Deck
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49
Find the area of the part of hyperbolic paraboloid that lies between the cylinders and .
A)
B)
C)
D)
E)
that lies between the cylinders and .A)
B)
C)
D)
E)
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50
Find the area of the surface. The part of the surface that lies above the xy-plane.
A)
B)
C)
D)
E)
that lies above the xy-plane.A)
B)
C)
D)
E)
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51
Express the triple integral as an iterated integral in six different ways using different orders of integration where T is the solid bounded by the planes and
as an iterated integral in six different ways using different orders of integration where T is the solid bounded by the planes and Unlock Deck
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52
Sketch the solid whose volume is given by the iterated integral
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53
Find the area of the surface. Round your answer to three decimal places.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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54
Find the area of the part of the plane that lies in the first octant.
that lies in the first octant. Unlock Deck
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55
Find the area of the surface S where S is the part of the sphere that lies inside the cylinder
that lies inside the cylinder Unlock Deck
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56
Sketch the solid bounded by the graphs of the equations and , and then use a triple integral to find the volume of the solid.
and , and then use a triple integral to find the volume of the solid. Unlock Deck
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57
Find the area of the part of the plane that lies inside the cylinder .
that lies inside the cylinder . Unlock Deck
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58
Find the area of the surface S where S is the part of the plane that lies above the triangular region with vertices , and
that lies above the triangular region with vertices , and Unlock Deck
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59
Find the exact area of the surface. .
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
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60
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 these
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 these
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61
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
and the x-axis. A)
B)
C)
D)
E)None of these
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62
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
and and having the mass density Unlock Deck
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63
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 = 2
P1(-3, -3), P2(0, 3), P3(-2, -1)
m1 = 4, m2 = 3, m3 = 2
P1(-3, -3), P2(0, 3), P3(-2, -1)
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64
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) ,
C) ,
D) ,
and , and having the mass density A)
, B)
, C)
, D)
, Unlock Deck
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65
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 .
A)
B)27
C)
D)
E)None of these
and the line . A)
B)27
C)
D)
E)None of these
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66
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 the equations and and having the mass density
and and having the mass density Unlock Deck
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67
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 rectangular region with vertices and , and having uniform density
and and the radii of gyration and for the lamina occupying the region R, where R is the rectangular region with vertices and , and having uniform density Unlock Deck
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68
Use polar coordinates to find the volume of the solid inside the cylinder and the ellipsoid .
A)
B)
C)
D)
E)
and the ellipsoid .A)
B)
C)
D)
E)
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69
A lamina occupies the part of the disk in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis.
in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis. Unlock Deck
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70
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.
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71
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
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 Unlock Deck
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72
Use polar coordinates to find the volume of the solid under the paraboloid and above the disk .
A)
B)
C)
D)
E)
and above the disk .A)
B)
C)
D)
E)
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73
Find the mass of the lamina that occupies the region and has the given density function. Round your answer to two decimal places.
and has the given density function. Round your answer to two decimal places. Unlock Deck
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74
Use polar coordinates to find the volume of the sphere of radius . Round to two decimal places.
A)
B)
C)
D)
E)
. Round to two decimal places.A)
B)
C)
D)
E)
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75
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)
-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)
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76
Use polar coordinates to find the volume of the solid bounded by the paraboloid and the plane .
A)
B)
C)
D)
E)
and the plane .A)
B)
C)
D)
E)
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77
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) coulombs
B) coulombs
C) coulombs
D) coulombs
Find the total charge on R if the charge density at a point in R (measured in coulombs per square meter) is A)
coulombsB)
coulombsC)
coulombsD)
coulombs Unlock Deck
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78
Use a double integral to find the area of the region R where R is bounded by the circle
A)
B)
C)
D)
A)
B)
C)
D)
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79
Evaluate the integral by changing to polar coordinates. is the region bounded by the semicircle and the -axis.
is the region bounded by the semicircle and the -axis. Unlock Deck
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80
Evaluate the iterated integral by converting to polar coordinates. Round the answer to two decimal places. .
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
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Unlock Deck
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