تكلم مع الذكاء الاصطناعي
Deck 15: Multiple Integrals
ملء الشاشة (f)
سؤال
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)
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
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
سؤال
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)
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 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
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
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
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
Identify the surface with equation 
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)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
سؤال
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. 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)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)
سؤال
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
سؤال
Find the Jacobian of the transformation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Find the Jacobian of the transformation. 
سؤال
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 
سؤال
Identify the surface with equation 
سؤال
Use spherical coordinate to find the volume above the cone 
and inside sphere 
.
and inside sphere . سؤال
Identify the surface with equation 
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
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. سؤال
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
سؤال
Find the region E for which the triple integral 
is a maximum.
is a maximum. سؤال
Evaluate the iterated integral 
سؤال
Use cylindrical coordinates to evaluate 
A)
B)
C)
D)
A)
B)
C)
D)
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)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
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 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
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
سؤال
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.
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
Find the center of mass of a homogeneous solid bounded by the paraboloid 
and 
and سؤال
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 . 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
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
سؤال
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
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
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
Evaluate the triple integral. Round your answer to one decimal place. 
سؤال
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 سؤال
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 سؤال
Sketch the solid whose volume is given by the integral 
Evaluate the integral.
Evaluate the integral. سؤال
Describe the region whose area is given by the integral. 
سؤال
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 سؤال
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)
سؤال
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)
سؤال
Find the area of the surface. The part of the surface 
that lies within the cylinder 
.
that lies within the cylinder . سؤال
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)
سؤال
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)
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
Sketch the solid whose volume is given by the iterated integral 
سؤال
Find the area of the surface. Round your answer to three decimal places. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Find the area of the part of the plane 
that lies in the first octant.
that lies in the first octant. سؤال
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 سؤال
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. سؤال
Find the area of the part of the plane 
that lies inside the cylinder 
.
that lies inside the cylinder . سؤال
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 سؤال
Find the exact area of the surface. 
.
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
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
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
سؤال
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 سؤال
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
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
سؤال
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 سؤال
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 سؤال
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)
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
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. 
سؤال
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 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
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
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
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
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
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 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
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. 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)
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العب
ملء الشاشة (f)
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. فتح الحزمة
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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 فتح الحزمة
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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. فتح الحزمة
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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 . فتح الحزمة
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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 . فتح الحزمة
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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. فتح الحزمة
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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. فتح الحزمة
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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 . فتح الحزمة
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33
Find the center of mass of a homogeneous solid bounded by the paraboloid and
and فتح الحزمة
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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 . فتح الحزمة
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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 فتح الحزمة
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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 . فتح الحزمة
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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. فتح الحزمة
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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 فتح الحزمة
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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 فتح الحزمة
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43
Sketch the solid whose volume is given by the integral Evaluate the integral.
Evaluate the integral. فتح الحزمة
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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 فتح الحزمة
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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 . فتح الحزمة
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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 فتح الحزمة
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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. فتح الحزمة
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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 فتح الحزمة
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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. فتح الحزمة
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57
Find the area of the part of the plane that lies inside the cylinder .
that lies inside the cylinder . فتح الحزمة
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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 فتح الحزمة
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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 فتح الحزمة
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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)
, فتح الحزمة
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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 فتح الحزمة
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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 فتح الحزمة
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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. فتح الحزمة
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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 فتح الحزمة
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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. فتح الحزمة
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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 فتح الحزمة
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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. فتح الحزمة
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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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