Deck 14: Multiple Integrals

Question 1

Find an integral equal to the volume of the solid bounded by the given surfaces and evaluate the integral.

Answer 1

Question 2

Evaluate the iterated integral by first changing the order of integration.

A)
B)
C)
D) The integral can be evaluated only numerically.

Answer 2

C

Question 3

Find the volume of the solid bounded by the given surfaces.

A)
B)
C) 40
D)

Answer 3

D

Question 4

Approximate the double integral. , where R is bounded by x = 0, x = 3, y = 0, and y = x + 2

A) 2.636
B) 2.658
C) 0.886
D) 1.772

Question 5

Compute the volume of the solid bounded by the given surfaces. and the three coordinate planes

A)
B)
C)
D)

Question 6

Evaluate the double integral.

A)
B)
C)
D)

Question 7

Evaluate the iterated integral by first changing the order of integration.

A)
B)
C)
D) The integral can be evaluated only numerically.

Question 8

Use a double integral to find the area of the region bounded by , and .

A)
B)
C)
D)

Question 9

Evaluate the iterated integral by first changing the order of integration.

A)
B)
C)
D)

Question 10

Evaluate the double integral.

A) 208
B) 304
C) -208
D) -304

Question 11

Compute the Riemann sum for the given function and region, a partition with n equal-sized rectangles and the given evaluation rule.

A) -8
B) 8
C) 0
D) -36

Question 12

Compute the Riemann sum for the given function, the irregular partition shown and midpoint evaluation.

A) 69
B) 40
C) -66
D) 66

Question 13

Change the order of integration.

A)
B)
C)
D)

Question 14

Change the order of integration.

A)
B)
C)
D)

Question 15

Evaluate the iterated integral by first changing the order of integration.

A)
B)
C)
D)

Question 16

Evaluate the iterated integral.

A)
B)
C)
D)

Question 17

Find the volume of the solid bounded by the given surfaces.

A)
B)
C) 40
D) 52

Question 18

Evaluate the iterated integral.

A)
B)
C)
D)

Question 19

Evaluate the iterated integral.

A)
B)
C)
D)

Question 20

Use a double integral to find the area of the region bounded by and .

A)
B)
C)
D)

Question 21

Find the mass and moments of inertia I_x and I_y for a lamina in the shape of the region bounded by and with density .

A)
B)
C)
D)

Question 22

Use polar coordinates to evaluate where R is the disk .

A)
B)
C)
D)

Question 23

A skydiving club is having a competition to see who can land the closest to a target point. Jeff is a highly experienced skydiver, and the probability that he will land inside a region R is given by , where the coordinate system is centered on the target point. Compute the probability that Jeff lands within 11 feet of the target point.

A)
B)
C)
D)

Question 24

Find the mass and the center of mass of the lamina bounded by and with density .

A)
B)
C)
D)

Question 25

Use an appropriate coordinate system to compute the volume of the solid below and above .

A)
B)
C)
D)

Question 26

Find the surface area of the portion of the surface in the first octant.

A)
B)
C)
D)

Question 27

Find a constant c such that is a joint pdf on the region bounded by and

A)
B)
C)
D)

Question 28

Find the center of mass of a lamina in the shape of , with density

A)
B)
C)
D)

Question 29

Use an appropriate coordinate system to compute the volume of the solid below , above and inside .

A)
B)
C)
D)

Question 30

Suppose that is the population density of a species of a certain small animal. Estimate the population in the triangular region and .

A) 2426
B) 1115
C) 2548
D) 936

Question 31

Find the surface area of the portion of below .

A)
B)
C)
D)

Question 32

Evaluate by converting to polar coordinates.

A)
B)
C)
D)

Question 33

Compute the volume of the solid bounded by , , , and .

A)
B)
C)
D)

Question 34

Find the area of the region bounded by

A)
B)
C)
D)

Question 35

Find the surface area of the portion of above the xy plane.

A)
B)
C)
D)

Question 36

Evaluate the iterated integral by converting to polar coordinates.

A)
B)
C)
D)

Question 37

Compute the volume of the solid bounded by , and the coordinate planes.

A)
B)
C)
D)

Question 38

Find the surface area of the portion of above .

A)
B)
C)
D)

Question 39

Find the surface area of between , and .

A)
B)
C)
D)

Question 40

Use polar coordinates to evaluate where R is the disk .

A)
B)
C)
D)

Question 41

Find the center of mass of the solid with density and the given shape. ,

A)
B)
C)
D)

Question 42

Find the mass of the solid with density and the given shape.

A)
B)
C)
D)

Question 43

Evaluate the integral , where Q is the region with z > 0 bounded by and .

A)
B)
C)
D)

Question 44

A function is a pdf on the three-dimensional region Q if for all in Q and . Find k such that is a pdf on the region bounded by z = 0, z = 3, x = 0, x = 2, y - z = 0 , and y - z = 3.

A)
B)
C)
D)

Question 45

Evaluate the triple integral.

A)
B)
C)
D)

Question 46

Evaluate the triple integral . ,

A)
B) 2
C)
D)

Question 47

Evaluate the triple integral . ,

A) 8
B)
C) 0
D)

Question 48

Write the equation in cylindrical coordinates.

A)
B)
C)
D)

Question 49

Set up the triple integral in cylindrical coordinates where Q is the solid above below and inside

A)
B)
C)
D)

Question 50

Find the center of mass of the solid with density and the given shape.

A)
B)
C)
D)

Question 51

Compute the volume of the solid bounded by the given surfaces.

A)
B)
C)
D)

Question 52

Rewrite the iterated integral by iterating in the order

A)
B)
C)
D)

Question 53

Evaluate the integral , where Q is the region with z > 0 bounded by and .

A)
B)
C)
D)

Question 54

Numerically estimate the surface area of the portion of inside of

A) 3.14
B) 5.57
C) 116.24
D) 114.51

Question 55

Evaluate the triple integral . ,

A)
B)
C)
D)

Question 56

Compute the volume of the solid bounded by the given surfaces.

A)
B)
C)
D)

Question 57

Which if the following could represent the triple integral in cylindrical coordinates where Q is the region below and above ?

A)
B)
C)
D)

Question 58

Write the equation in cylindrical coordinates.

A)
B)
C)
D)

Question 59

Which of the following could represent the triple integral in cylindrical coordinates where Q is the region bounded below by and above by ?

A)
B)
C)
D)

Question 60

Find the mass of the solid with density and the given shape. ,

A)
B)
C)
D)

Question 61

Evaluate the iterated integral after changing coordinate systems.

A)
B)
C)
D)

Question 62

Set up and evaluate the integral where Q is the region bounded by the coordinate planes and the planes x = 2, y = 2, and z = 2.

A)
B)
C)
D)

Question 63

Convert the point to rectangular coordinates (x,y,z).

A)
B)
C)
D)

Question 64

Evaluate the triple integral where Q is the bounded by and

A)
B)
C)
D)

Question 65

Convert the point to rectangular coordinates (x,y,z).

A)
B)
C)
D)

Question 66

Using an appropriate coordinate system, evaluate the integral where Q is the region above z = 0 bounded by the cone and the sphere .

A)
B)
C)
D)

Question 67

Calculate the mass of an object with density and bounded by and the planes .

A)
B)
C)
D)

Question 68

Calculate the mass of an object with density and bounded by and

A)
B)
C)
D)

Question 69

Using an appropriate coordinate system, evaluate the integral where Q is the region inside the cylinder and between the planes and .

A)
B)
C)
D)

Question 70

Use an appropriate coordinate system to find the volume of a solid lying outside the cones defined by (includes the portion extending to z < 0) and inside the sphere defined by .

A)
B)
C)
D)

Question 71

Evaluate the iterated integral by changing coordinate systems.

A)
B)
C)
D)

Question 72

Convert the equation in spherical coordinates to an equation in rectangular coordinates.

A)
B)
C)
D)

Question 73

Set up and evaluate the integral where Q is the region above the xy-plane bounded by the hemisphere centered at (0,0,0) and with a radius of 5.

A)
B)
C) 0
D)

Question 74

Convert the equation into spherical coordinates.

A)
B)
C)
D)

Question 75

Set up and evaluate the integral where Q is the region inside a sphere centered at (0,0,0) and with a radius of 7.

A)
B) 0
C)
D)

Question 76

Use an appropriate coordinate system to find the volume of the solid lying along the positive z-axis and bounded by the cone and the sphere .

A)
B)
C)
D)

Question 77

Compute the volume of the solid Q bounded by and .

A)
B)
C)
D)

Question 78

Evaluate the iterated integral after changing coordinate systems.

A)
B)
C)
D)

Question 79

Using an appropriate coordinate system, evaluate the integral where Q is the region inside the cylinder and between the planes and .

A)
B)
C)
D)

Question 80

Evaluate the iterated integral by changing coordinate systems.

A)
B)
C)
D)