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Deck 12: Multiple Integrals
ملء الشاشة (f)
سؤال
Find the center of mass of a homogeneous solid bounded by the paraboloid 
and 
and 
سؤال
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)175.37
C)176.38
D)175.93
E)
where E lies above the paraboloid and below the plane .A)
160.28B)175.37
C)176.38
D)175.93
E)
سؤال
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)
سؤال
Find the mass of the solid S bounded by the paraboloid 
and the plane 
if S has constant density 3.
A)15.07
B)16.25
C)24.91
D)13.92
E)19.63
and the plane if S has constant density 3.A)15.07
B)16.25
C)24.91
D)13.92
E)19.63
سؤال
Find the Jacobian of the transformation. 
سؤال
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)0.54
B)0
C)3.4
D)8.57
E)9.19
where E is the solid that lies between the cylinders and above the xy-plane and below the plane .A)0.54
B)0
C)3.4
D)8.57
E)9.19
سؤال
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)
سؤال
Find the Jacobian of the transformation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
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)4.447
B)3.296
C)5.088
D)9.447
E)8.841
, where R is the region in the first quadrant bounded by the lines and the hyperbolas .A)4.447
B)3.296
C)5.088
D)9.447
E)8.841
سؤال
Identify the surface with equation 
سؤال
Identify the surface with equation 
سؤال
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)
سؤال
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 سؤال
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 . سؤال
Calculate the iterated integral. 
A)
B)8
C)
D)
E)None of these
A)
B)8
C)
D)
E)None of these
سؤال
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. سؤال
Use the given transformation to evaluate the integral. 
, where R is the square with vertices (0, 0), (4, 6), (6, 
), (10, 2) and 
A)208
B)42
C)312
D)52
E)343
, where R is the square with vertices (0, 0), (4, 6), (6, ), (10, 2) and A)208
B)42
C)312
D)52
E)343
سؤال
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)
C)205.13
D)198.08
E)213.5
and density 1 about a diameter of its base.A)195.22
B)
C)205.13
D)198.08
E)213.5
سؤال
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
سؤال
Use cylindrical coordinates to evaluate 
A)
B)
C)
D)
A)
B)
C)
D)
سؤال
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 سؤال
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)
سؤال
Evaluate the triple integral. Round your answer to one decimal place. 
سؤال
Find the mass of the solid E, if E is the cube given by 
and the density function 
is 
.
and the density function is . سؤال
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)
سؤال
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 . سؤال
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. سؤال
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)
, سؤال
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. سؤال
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 سؤال
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 . سؤال
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)
سؤال
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. سؤال
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)
سؤال
Sketch the solid whose volume is given by the iterated integral 
سؤال
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)
سؤال
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)
سؤال
A cylindrical drill with radius 
is used to bore a hole through the center of a sphere of radius 
. Find the volume of the ring-shaped solid that remains. Round the answer to the nearest hundredth.
is used to bore a hole through the center of a sphere of radius . Find the volume of the ring-shaped solid that remains. Round the answer to the nearest hundredth. سؤال
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
سؤال
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 سؤال
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. 
, where f is a continuous function. Then write an expression for the (iterated) integral. سؤال
Evaluate the integral by reversing the order of integration. 
سؤال
Evaluate the double integral. 
, 
is triangular region with vertices 
.
, is triangular region with vertices . سؤال
The double integral 
, where 
, gives the volume of a solid. Describe the solid.
, where , gives the volume of a solid. Describe the solid. سؤال
Use polar coordinates to evaluate. 
سؤال
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.
A)
B)
C)
D)
E)
and below the elliptic paraboloid .Divide into four equal squares and use the Midpoint rule.A)
B)
C)
D)
E)
سؤال
Evaluate 
where 
is the figure bounded by 
and 
.
where is the figure bounded by and . سؤال
Evaluate the double integral by first identifying it as the volume of a solid. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Evaluate the double integral. 
is bounded by 
and 
.
is bounded by and . سؤال
Evaluate the integral 
, where R is the annular region bounded by the circles 
and 
by changing to polar coordinates.
, where R is the annular region bounded by the circles and by changing to polar coordinates. سؤال
Evaluate the double integral. 
is bounded by the circle with center the origin and radius 
.
is bounded by the circle with center the origin and radius . سؤال
Find the volume under 
and above the region bounded by 
and 
.
A)
B)
C)
D)
E)
and above the region bounded by and .A)
B)
C)
D)
E)
سؤال
Calculate the double integral. Round your answer to two decimal places. 
سؤال
Calculate the double integral. Round your answer to two decimal places. 
سؤال
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?
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? سؤال
Evaluate the integral by reversing the order of integration. 
سؤال
Calculate the iterated integral. 
سؤال
Evaluate the double integral 
, where 
is the region bounded by the graphs of 
and 
.
, where is the region bounded by the graphs of and . سؤال
Evaluate the iterated integral 
by reversing the order of integration.
by reversing the order of integration.
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ملء الشاشة (f)
Deck 12: Multiple Integrals
1
Find the center of mass of a homogeneous solid bounded by the paraboloid and
and 
2
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)175.37
C)176.38
D)175.93
E)
where E lies above the paraboloid and below the plane .A)
160.28B)175.37
C)176.38
D)175.93
E)
3
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)
4
Find the mass of the solid S bounded by the paraboloid and the plane if S has constant density 3.
A)15.07
B)16.25
C)24.91
D)13.92
E)19.63
and the plane if S has constant density 3.A)15.07
B)16.25
C)24.91
D)13.92
E)19.63
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5
Find the Jacobian of the transformation.
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6
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)0.54
B)0
C)3.4
D)8.57
E)9.19
where E is the solid that lies between the cylinders and above the xy-plane and below the plane .A)0.54
B)0
C)3.4
D)8.57
E)9.19
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7
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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8
Find the Jacobian of the transformation.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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9
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)4.447
B)3.296
C)5.088
D)9.447
E)8.841
, where R is the region in the first quadrant bounded by the lines and the hyperbolas .A)4.447
B)3.296
C)5.088
D)9.447
E)8.841
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10
Identify the surface with equation
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11
Identify the surface with equation
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12
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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13
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 فتح الحزمة
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14
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 . فتح الحزمة
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15
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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16
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. فتح الحزمة
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17
Use the given transformation to evaluate the integral. , where R is the square with vertices (0, 0), (4, 6), (6, ), (10, 2) and
A)208
B)42
C)312
D)52
E)343
, where R is the square with vertices (0, 0), (4, 6), (6, ), (10, 2) and A)208
B)42
C)312
D)52
E)343
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18
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)
C)205.13
D)198.08
E)213.5
and density 1 about a diameter of its base.A)195.22
B)
C)205.13
D)198.08
E)213.5
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19
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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20
Use cylindrical coordinates to evaluate
A)
B)
C)
D)
A)
B)
C)
D)
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21
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 فتح الحزمة
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22
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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23
Evaluate the triple integral. Round your answer to one decimal place.
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24
Find the mass of the solid E, if E is the cube given by and the density function is .
and the density function is . فتح الحزمة
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25
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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26
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 . فتح الحزمة
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27
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. فتح الحزمة
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28
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)
, فتح الحزمة
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29
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. فتح الحزمة
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30
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 فتح الحزمة
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31
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 . فتح الحزمة
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32
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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33
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. فتح الحزمة
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34
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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35
Sketch the solid whose volume is given by the iterated integral
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36
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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37
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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38
A cylindrical drill with radius is used to bore a hole through the center of a sphere of radius . Find the volume of the ring-shaped solid that remains. Round the answer to the nearest hundredth.
is used to bore a hole through the center of a sphere of radius . Find the volume of the ring-shaped solid that remains. Round the answer to the nearest hundredth. فتح الحزمة
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39
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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40
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 فتح الحزمة
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41
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.
, where f is a continuous function. Then write an expression for the (iterated) integral. فتح الحزمة
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42
Evaluate the integral by reversing the order of integration.
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43
Evaluate the double integral. , is triangular region with vertices .
, is triangular region with vertices . فتح الحزمة
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44
The double integral , where , gives the volume of a solid. Describe the solid.
, where , gives the volume of a solid. Describe the solid. فتح الحزمة
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45
Use polar coordinates to evaluate.
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46
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.
A)
B)
C)
D)
E)
and below the elliptic paraboloid .Divide into four equal squares and use the Midpoint rule.A)
B)
C)
D)
E)
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47
Evaluate where is the figure bounded by and .
where is the figure bounded by and . فتح الحزمة
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48
Evaluate the double integral by first identifying it as the volume of a solid.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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49
Evaluate the double integral. is bounded by and .
is bounded by and . فتح الحزمة
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50
Evaluate the integral , where R is the annular region bounded by the circles and by changing to polar coordinates.
, where R is the annular region bounded by the circles and by changing to polar coordinates. فتح الحزمة
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51
Evaluate the double integral. is bounded by the circle with center the origin and radius .
is bounded by the circle with center the origin and radius . فتح الحزمة
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52
Find the volume under and above the region bounded by and .
A)
B)
C)
D)
E)
and above the region bounded by and .A)
B)
C)
D)
E)
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53
Calculate the double integral. Round your answer to two decimal places.
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54
Calculate the double integral. Round your answer to two decimal places.
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55
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?
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? فتح الحزمة
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56
Evaluate the integral by reversing the order of integration.
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57
Calculate the iterated integral.
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58
Evaluate the double integral , where is the region bounded by the graphs of and .
, where is the region bounded by the graphs of and . فتح الحزمة
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59
Evaluate the iterated integral by reversing the order of integration.
by reversing the order of integration. فتح الحزمة
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