Deck 15: Multiple Integrals

Question 1

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

A)
B)
C)
D)

Answer 1

Question 2

Evaluate the double integral.

A)
B)
C)
D)

Answer 2

Question 3

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

A)
B)
C) 40
D)

Answer 3

Question 4

Approximate the double integral. , where R is bounded by x = 0, x = 1, y = 0, and y = 2x + 3

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

Question 5

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

A)
B)
C)
D)

Question 6

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

A) 48
B) 33
C) -57
D) 57

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

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 9

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

Question 10

Change the order of integration.

A)
B)
C)
D)

Question 11

Evaluate the iterated integral.

A)
B)
C)
D)

Question 12

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

A)
B)
C)
D)

Question 13

Change the order of integration.

A)
B)
C)
D)

Question 14

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

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

Question 15

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

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) 117
D) 351

Question 18

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

A)
B)
C)
D)

Question 19

Evaluate the iterated integral.

A)
B)
C)
D)

Question 20

Evaluate the double integral.

A) 352
B) 416
C) -352
D) -416

Question 21

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

A)
B)
C)
D)

Question 22

Find the surface area of between , and .

A)
B)
C)
D)

Question 23

Evaluate by converting to polar coordinates.

A)
B)
C)
D)

Question 24

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

A) 4578
B) 1916
C) 4704
D) 1740

Question 25

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

A)
B)
C)
D)

Question 26

Find the surface area of the portion of above .

A)
B)
C)
D)

Question 27

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

A)
B)
C)
D)

Question 28

Evaluate the iterated integral by converting to polar coordinates.

A)
B)
C)
D)

Question 29

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

A)
B)
C)
D)

Question 30

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 31

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

A)
B)
C)
D)

Question 32

Find the area of the region bounded by

A)
B)
C)
D)

Question 33

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

A)
B)
C)
D)

Question 34

Use polar coordinates to evaluate where R is the disk .

A)
B)
C)
D)

Question 35

Use polar coordinates to evaluate where R is the disk .

A)
B)
C)
D)

Question 36

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 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 below .

A)
B)
C)
D)

Question 39

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

A)
B)
C)
D)

Question 40

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

A)
B)
C)
D)

Question 41

Evaluate the triple integral . ,

A)
B)
C)
D)

Question 42

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

A)
B)
C)
D)

Question 43

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

A)
B)
C)
D)

Question 44

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

A)
B)
C)
D)

Question 45

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

A)
B)
C)
D)

Question 46

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

A)
B)
C)
D)

Question 47

Write the equation in cylindrical coordinates.

A)
B)
C)
D)

Question 48

Numerically estimate the surface area of the portion of inside of

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

Question 49

Evaluate the triple integral.

A)
B)
C)
D)

Question 50

Write the equation in cylindrical coordinates.

A)
B)
C)
D)

Question 51

Evaluate the triple integral . ,

A) 4
B)
C) 0
D)

Question 52

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

A)
B)
C)
D)

Question 53

Evaluate the triple integral . ,

A)
B) 10
C)
D)

Question 54

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

A)
B)
C)
D)

Question 55

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

A)
B)
C)
D)

Question 56

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 = 1, y - z = 0 , and y - z = 3.

A)
B)
C)
D)

Question 57

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

A)
B)
C)
D)

Question 58

Rewrite the iterated integral by iterating in the order

A)
B)
C)
D)

Question 59

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 60

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 61

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

A)
B)
C)
D)

Question 62

Evaluate the iterated integral by changing coordinate systems.

A)
B)
C)
D)

Question 63

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 64

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 65

Evaluate the iterated integral after changing coordinate systems.

A)
B)
C)
D)

Question 66

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

A)
B)
C)
D)

Question 67

Compute the volume of the solid Q bounded by and .

A)
B)
C)
D)

Question 68

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 69

Evaluate the triple integral where Q is the bounded by and

A)
B)
C)
D)

Question 70

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 71

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

A)
B)
C)
D)

Question 72

Convert the equation into spherical coordinates.

A)
B)
C)
D)

Question 73

Evaluate the iterated integral by changing coordinate systems.

A)
B)
C)
D)

Question 74

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 4.

A)
B)
C) 0
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 5.

A)
B) 0
C)
D)

Question 76

Evaluate the iterated integral after changing coordinate systems.

A)
B)
C)
D)

Question 77

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 78

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

A)
B)
C)
D)

Question 79

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

A)
B)
C)
D)

Question 80

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)