Deck 16: Multiple Integration

Question 1

Find the volume of the solid which lies under the surface and above the rectangle .

Answer 1

Question 2

The value of the integral is which of the following?

A)
B)
C)
D)
E) None of the above.

Answer 2

Question 3

Evaluate , where is the rectangle .

Answer 3

Question 4

Compute the integral .

Question 5

Find the volume of the region under the surface and above the rectangle

Question 6

Evaluate the integral .

Question 7

Evaluate the double integral of the function over the rectangle

Question 8

Evaluate the iterated integral .

Question 9

Let be the following function: .
a) Compute and .
b) Is Fubini's Theorem valid in this case? If not, explain why.

Question 10

Evaluate .

Question 11

Sketch the region of integration and compute ,
where is the region enclosed by the parabolas and .

Question 12

Evaluate the double integral of the function over the rectangle.

A)
B)

Question 13

Compute .

Question 14

Let be continuous. If the equality holds, then:

A) .
B) .
C) .
D) .
E) There are no , and that satisfy the equality.

Question 15

Compute .

Question 16

Evaluate .

Question 17

Find the volume of the solid bounded by the surface and the planes , and

Question 18

Compute , where is the parallelogram with edges , and .

Question 19

Find the volume of the region under the surface and above the rectangle

Question 20

The value of the integral is which of the following?

A)
B)
C)
D) 1
E) None of the above.

Question 21

Evaluate , where is the triangle with vertices and .

Question 22

Let .

A) Rewrite the integral in the order
B) Compute the integral in the preferable order.

Question 23

The volume of the tetrahedron bounded by the coordinate planes and the plane through and is which of the following?

A)
B)
C)
D)
E)

Question 24

Evaluate where is the region enclosed by the planes , and .

Question 25

A lamina bounded by the curves , , and has mass density . Find the mass of the lamina.

Question 26

The value of the integral is which of the following?

A)
B)
C)
D)
E) The integral cannot be evaluated analytically.

Question 27

Evaluate the , where is the region .
Sketch the region of integration.

Question 28

The volume of the tetrahedron with the vertices and is which of the following?

A)
B)
C)
D)
E) None of the above.

Question 29

Given the integral :
a) sketch the region of integration and reverse the order of integration.
b) evaluate the integral.

Question 30

Compute the .
Sketch the region of integration.

Question 31

Let and be the triangle in the plane enclosed by the lines .
Find the set of all the points in such that

Question 32

a) Change the order of integration in the integral .
b) Evaluate the integral.

Question 33

Let be the following function on the rectangle : .
A) ?

A) Compute
B) Can you find in such that is equal to the value in
C) Explain why there is no contradiction with the Mean Value Theorem.

Question 34

Consider the integral

A) Draw the region of integration and reverse the order of integration.
B) Compute the integral in any order you choose.

Question 35

Rewrite the following integral in the order .

Question 36

Evaluate the triple integral where is the solid tetrahedron with vertices and .

Question 37

Consider the integral .

A) Sketch the region of integration
B) Interchange the order of integration.
C) Evaluate the integral.

Question 38

Let a) Sketch the region of integration and reverse the order of integration.
b) Compute the integral for .

Question 39

a) Change the order of integration in the integral .
b) Evaluate the integral.

Question 40

The volume of the ellipsoid is which of the following?

A)
B)
C)
D)
E)

Question 41

Integrate over the region in the first octant above and below

Question 42

Compute the volume of the region enclosed by the plane and the paraboloid .

Question 43

Compute the volume of the region satisfying and

Question 44

Let , , and Which of the following statements holds?

A) .
B) but .
C) but .
D) but .
E) The three integrals are different.

Question 45

Use cylindrical coordinates to compute the volume of the solid enclosed by the surfaces and .

Question 46

Use polar coordinates to compute , where is the region in the first quadrant enclosed by the circles and .

Question 47

Let where is the tetrahedron with vertices at , , and Write as an iterated integral in the orders and .

Question 48

Use cylindrical coordinates to compute the volume of the solid bounded by the paraboloid and the upper sphere .

Question 49

Find the volume of the ellipsoid

Question 50

The average value of over the tetrahedron with vertices and is which of the following?

A)
B)
C)
D)
E)

Question 51

Evaluate where is the tetrahedron with vertices at , , and in the order .

Question 52

Rewrite the integral in the order .

Question 53

Convert the following integral to cylindrical and spherical coordinates: .

Question 54

The value of is which of the following?

A)
B)
C)
D)
E) None of these answers are correct.

Question 55

Rewrite the integral using cylindrical coordinates and evaluate the integral.

Question 56

Let be the region Using polar coordinates for the integral

A) find the limits of .
B) find the limits of .
C) convert the integral to polar coordinates.

Question 57

Find if is the region satisfying and

Question 58

Let be the region in the space defined by the inequalities

A) Rewrite the inequalities for in adequate form for evaluating in the order .
B) Evaluate the integral .

Question 59

The volume of the region enclosed by the cylinder , the cone and the plane is which of the following?

A)
B)
C)
D)
E) None of the above.

Question 60

Use spherical coordinates to compute , where is the solid bounded by the spheres and .

Question 61

Find the coordinates of the centroid of the plate bounded by the lines and

Question 62

The solid in the first octant bounded by the sphere above, by the cone below, and by the planes and on the side, has a density .
Set up the integral for the mass of the solid

A) in rectangular coordinates.
B) in cylindrical coordinates.
C) in spherical coordinates.

Question 63

Find the coordinates of the centroid for the sector of the unit disk satisfying where

Question 64

Find the volume of the portion of the ball lying above the plane

Question 65

Integrate the function over the region bounded by and

Question 66

Let be the solid occupying the region Find the moment of inertia about the axis if has density

Question 67

Integrate the function over the region bounded by the cone and the paraboloid

Question 68

Use spherical coordinates to set up the integral to compute the volume of the solid bounded by the paraboloid and the upper sphere . (Do not evaluate the integral.)

Question 69

If the density is find the mass of the solid in the first octant bounded by the sphere above, by the cone below, and by the planes and on the side.

A)
B)
C)
D)
E) None of the above

Question 70

If the mass density is find for the solid occupying the region which satisfies and

Question 71

Find the mass and of the solid in the first octant bounded by the cylinder and the planes and , assuming that the mass density is .

Question 72

Find the coordinates of the centroid for the region described by

Question 73

The of the solid region that is inside the cylinder , below the paraboloid and above the plane (assuming ) is which of the following?

A)
B)
C)
D) 6
E) None of the above

Question 74

Find the volume of the region inside the cylinder but outside the paraboloid

Question 75

Use cylindrical coordinates to compute the volume of the solid .

Question 76

Find the volume of the region satisfying and

Question 77

Find the volume of the portion of the ball satisfying and lying above the surface defined by

Question 78

Find the center of mass of the tetrahedron in the first octant formed by the coordinate planes and the plane . (Assume the density is .)

Question 79

Consider a flat plate occupying the region in the plane bounded by the curves and If the plate has density find the moment of inertia about the axis.

Question 80

Let be the solid bounded by the cylinder and the planes , and in the first octant. The density of the solid is .

A) Find the mass of the solid.
B) Find the center of mass of the solid.