Deck 12: Multiple Integrals

Question 1

Find the center of mass of a homogeneous solid bounded by the paraboloid and

Answer 1

Question 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)

Answer 2

Question 3

Use the transformation to evaluate the integral , where R is the region bounded by the ellipse .

A)
B)
C)
D)
E)

Answer 3

Question 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

Question 5

Find the Jacobian of the transformation.

Question 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

Question 7

Use a triple integral to find the volume of the solid bounded by and the planes and .

A)
B)
C)
D)
E)

Question 8

Find the Jacobian of the transformation.

A)
B)
C)
D)
E)

Question 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

Question 10

Identify the surface with equation

Question 11

Identify the surface with equation

Question 12

Use cylindrical coordinates to evaluate where T is the solid bounded by the cylinder and the planes and

A)
B)
C)
D)

Question 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

Question 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 .

Question 15

Calculate the iterated integral.

A)
B)8
C)
D)
E)None of these

Question 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.

Question 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

Question 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

Question 19

Use spherical coordinates.Evaluate , where is the ball with center the origin and radius .

A)
B)
C)
D)
E)None of these

Question 20

Use cylindrical coordinates to evaluate

A)
B)
C)
D)

Question 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

Question 22

Use polar coordinates to find the volume of the sphere of radius . Round to two decimal places.

A)
B)
C)
D)
E)

Question 23

Evaluate the triple integral. Round your answer to one decimal place.

Question 24

Find the mass of the solid E, if E is the cube given by and the density function is .

Question 25

Evaluate the iterated integral by converting to polar coordinates. Round the answer to two decimal places. .

A)
B)
C)
D)
E)

Question 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 .

Question 27

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

Question 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) ,

Question 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.

Question 30

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

Question 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 .

Question 32

Use a double integral to find the area of the region R where R is bounded by the circle

A)
B)
C)
D)

Question 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.

Question 34

Use polar coordinates to find the volume of the solid under the paraboloid and above the disk .

A)
B)
C)
D)
E)

Question 35

Sketch the solid whose volume is given by the iterated integral

Question 36

Use polar coordinates to find the volume of the solid bounded by the paraboloid and the plane .

A)
B)
C)
D)
E)

Question 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)

Question 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.

Question 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

Question 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

Question 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.

Question 42

Evaluate the integral by reversing the order of integration.

Question 43

Evaluate the double integral. , is triangular region with vertices .

Question 44

The double integral , where , gives the volume of a solid. Describe the solid.

Question 45

Use polar coordinates to evaluate.

Question 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)

Question 47

Evaluate where is the figure bounded by and .

Question 48

Evaluate the double integral by first identifying it as the volume of a solid.

A)
B)
C)
D)
E)

Question 49

Evaluate the double integral. is bounded by and .

Question 50

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

Question 51

Evaluate the double integral. is bounded by the circle with center the origin and radius .

Question 52

Find the volume under and above the region bounded by and .

A)
B)
C)
D)
E)

Question 53

Calculate the double integral. Round your answer to two decimal places.

Question 54

Calculate the double integral. Round your answer to two decimal places.

Question 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?

Question 56

Evaluate the integral by reversing the order of integration.

Question 57

Calculate the iterated integral.

Question 58

Evaluate the double integral , where is the region bounded by the graphs of and .

Question 59

Evaluate the iterated integral by reversing the order of integration.