# Quiz 12: Multiple Integrals

Mathematics

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Multiple Choice

E

Q 2Q 2

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

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C

Q 3Q 3

Use the transformation to evaluate the integral , where R is the region bounded by the ellipse .
A)
B)
C)
D)
E)

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C

Q 4Q 4

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

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Q 6Q 6

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 .

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Q 7Q 7

Use spherical coordinates.Evaluate , where is the ball with center the origin and radius .
A)
B)
C)
D)
E)None of these

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Q 8Q 8

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

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Q 9Q 9

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

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Q 10Q 10

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.

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Q 13Q 13

Use cylindrical coordinates to evaluate where T is the solid bounded by the cylinder and the planes and
A)
B)
C)
D)

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Q 14Q 14

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)

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Q 15Q 15

Use cylindrical coordinates to evaluate the triple integral where E is the solid that lies between the cylinders and above the xy-plane and below the plane .
A)0.54
B)0
C)3.4
D)8.57
E)9.19

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Q 19Q 19

Use a triple integral to find the volume of the solid bounded by and the planes and .
A)
B)
C)
D)
E)

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Q 20Q 20

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

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Q 22Q 22

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 .

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Q 23Q 23

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 .

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Q 24Q 24

Express the integral as an iterated integral of the form where E is the solid bounded by the surfaces

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Q 26Q 26

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.

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Q 28Q 28

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

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Q 29Q 29

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

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Multiple Choice

Q 30Q 30

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) ,

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Q 31Q 31

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

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Q 32Q 32

Find the mass of the lamina that occupies the region and has the given density function. Round your answer to two decimal places.

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Q 33Q 33

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.

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Q 34Q 34

Use polar coordinates to find the volume of the solid under the paraboloid and above the disk .
A)
B)
C)
D)
E)

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Q 35Q 35

Use polar coordinates to find the volume of the sphere of radius . Round to two decimal places.
A)
B)
C)
D)
E)

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Q 36Q 36

Use polar coordinates to find the volume of the solid bounded by the paraboloid and the plane .
A)
B)
C)
D)
E)

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Q 37Q 37

Use a double integral to find the area of the region R where R is bounded by the circle
A)
B)
C)
D)

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Q 38Q 38

Evaluate the iterated integral by converting to polar coordinates. Round the answer to two decimal places. .
A)
B)
C)
D)
E)

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Q 39Q 39

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)

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Q 40Q 40

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.

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Q 42Q 42

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?

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Q 43Q 43

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.

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Q 44Q 44

Evaluate the integral , where R is the annular region bounded by the circles and by changing to polar coordinates.

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Q 55Q 55

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)

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Q 56Q 56

Evaluate the double integral by first identifying it as the volume of a solid.
A)
B)
C)
D)
E)

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