Deck 15: Multiple Integrals

Question 1

Describe the region whose area is given by the integral.

Answer 1

half the region inside the loop of the four-leaved

Question 2

Find the area of the surface S where S is the part of the plane that lies above the triangular region with vertices

Answer 2

Question 3

Find the area of the surface S where S is the part of the sphere that lies inside the cylinder

Answer 3

Question 4

Determine whether to use polar coordinates or rectangular coordinates to evaluate the integral where is a continuous function.Then write an expression for the (iterated) integral.

Question 5

Find the center of mass of the lamina of the region shown if the density of the circular lamina is four times that of the rectangular lamina.

Question 6

Use spherical coordinate to find the volume above the cone and inside sphere

Question 7

Calculate the iterated integral.

Question 8

Find the mass and the moments of inertia and the radii of gyration for the lamina occupying the region R, where R is the region bounded by the graphs of the equations and having the mass density

Question 9

Sketch the solid bounded by the graphs of the equations and and then use a triple integral to find the volume of the solid.

Question 10

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

Question 11

Calculate the iterated integral.

Question 12

Evaluate the integral by changing to polar coordinates. is the region bounded by the semicircle and the -axis.

Question 13

Evaluate where is the figure bounded by

Question 14

Find the mass and the center of mass of the lamina occupying the region R, where R is the region bounded by the graphs of and having the mass density

Question 15

Sketch the solid whose volume is given by the iterated integral

Question 16

Calculate the iterated integral.

Question 17

Find the area of the surface.The part of the surface that lies within the cylinder

Question 18

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

Question 19

Find the volume of the solid bounded in the first octanat bounded by the cylinder and the planes

Question 20

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

Question 21

Calculate the iterated integral.

Question 22

Determine whether to use polar coordinates or rectangular coordinates to evaluate the integral where is a continuous function.Then write an expression for the (iterated) integral.

Question 23

Evaluate the triple integral.Round yourAnswer to one decimal place.

Question 24

Use polar coordinates to evaluate.

Question 25

Calculate the double integral.Round yourAnswer to two decimal places.

Question 26

Find the area of the surface S where S is the part of the surface that lies inside the cylinder

Question 27

Sketch the solid bounded by the graphs of the equations and and then use a triple integral to find the volume of the solid.

Question 28

Evaluate the integral by making an appropriate change of variables.Round yourAnswer to two decimal places. R is the parallelogram bounded by the lines

Question 29

Find the volume of the solid bounded in the first octanat bounded by the cylinder and the planes

Question 30

Identify the surface with equation

Question 31

Use spherical coordinate to find the volume above the cone and inside sphere

Question 32

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 33

Identify the surface with equation

Question 34

Evaluate the double integral

Question 35

Find the mass and the center of mass of the lamina occupying the region R, where R is the region bounded by the graphs of and having the mass density

Question 36

Evaluate the double integral is the region bounded by the graphs of

Question 37

Find the area of the surface S where S is the part of the sphere that lies to the right of the xz-plane and inside the cylinder

Question 38

Find the center of mass of the system comprising masses located at the points in a coordinate plane.Assume that mass is measured in grams and distance is measured in centimeters.

Question 39

Evaluate the integral by reversing the order of integration.

Question 40

Calculate the double integral.Round yourAnswer to two decimal places.

Question 41

Select the correct Answer for each question.
The sketch of the solid is given below.Given , write the inequalities that describe it.

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

Question 42

Select the correct Answer for each question.
Evaluate the iterated integral by converting to polar coordinates.Round theAnswer to two decimal places.

A)
B)
C)
D)
E)

Question 43

Select the correct Answer for each question.
Use spherical coordinates to evaluate where B is the ball

A)
B)
C)
D)

Question 44

Select the correct Answer for each question.
Calculate the iterated integral.

A)
B)4
C)
D)
E)None of these

Question 45

Select the correct Answer for each question.
Use cylindrical coordinates to evaluate the triple integral where E is the solid that lies between the cylinders above the xy-plane and below the plane

A)8.57
B)0
C)3.4
D)9.19
E)0.54

Question 46

Select the correct Answer for each question.
Use a triple integral to find the volume of the solid bounded by and the planes and

A)
B)
C)
D)
E)

Question 47

Select the correct Answer for each question.
Use polar coordinates to find the volume of the solid under the paraboloid and above the disk

A)
B)
C)
D)
E)

Question 48

Select the correct Answer for each question.
Use the Midpoint Rule with four squares of equal size to estimate the double integral.

A)
B)
C)
D)
E)

Question 49

Select the correct Answer for each question.
Use spherical coordinates.Evaluate here is the ball with center the origin and radius

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

Question 50

Select the correct Answer for each question.
Use cylindrical or spherical coordinates, whichever seems more appropriate, to evaluate where E lies above the paraboloid and below the plane

A)
B)
C)
D)
E)

Question 51

Select the correct Answer for each question.
Calculate the double integral.

A)
B)
C)
D)
E)

Question 52

Select the correct Answer for each question.
Find the area of the surface.The part of the surface that lies above the xy-plane.

A)
B)
C)
D)
E)

Question 53

Select the correct Answer for each question.
Evaluate the double integral by first identifying it as the volume of a solid.

A)
B)
C)
D)
E)

Question 54

Select the correct Answer for each question.
Find the Jacobian of the transformation.

A)
B)
C)
D)
E)

Question 55

Select the correct Answer for each question.
Use cylindrical coordinates to evaluate where T is the solid bounded by the cylinder and the planes and

A)
B)
C)
D)

Question 56

Select the correct Answer for each question.
Calculate the iterated integral.

A)
B)
C)
D)
E)

Question 57

Select the correct Answer for each question.
A swimming pool is circular with a 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 58

Select the correct Answer for each question.
Evaluate the iterated integral.

A)
B)
C)
D)
E)

Question 59

Select the correct Answer for each question.
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 60

Select the correct Answer for each question.
Use the transformation to evaluate the integral here R is the region bounded by the ellipse

A)
B)
C)
D)
E)

Question 61

Select the correct Answer for each question.
Calculate the double integral.

A)
B)
C)
D)
E)

Question 62

Select the correct Answer for each question.
An electric charge is spread over a rectangular region ind the total charge on R if the charge density at a point in R (measured in coulombs per square meter) is

A)84 coulombs
B)222 coulombs
C)22 coulombs
D)56 coulombs

Question 63

Select the correct Answer for each question.
Use polar coordinates to find the volume of the solid bounded by the paraboloid and the plane

A)
B)
C)
D)
E)

Question 64

Select the correct Answer for each question.
Use a computer algebra system to find the moment of inertia of the lamina that occupies the region D and has the density function

A)14.1
B)34.138
C)24.105
D)28.5
E)None of these

Question 65

Select the correct Answer for each question.
Find the center of mass of the lamina that occupies the region D and has the given density function, if D is bounded by the parabola and the x-axis.

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

Question 66

Select the correct Answer for each question.
Evaluate the double integral by first identifying it as the volume of a solid.

A)
B)
C)
D)
E)

Question 67

Select the correct Answer for each question.
Use polar coordinates to find the volume of the solid under the paraboloid and above the disk

A)
B)
C)
D)
E)

Question 68

Select the correct Answer for each question.
Calculate the iterated integral.Round yourAnswer to two decimal places.

A)
B)
C)
D)
E)

Question 69

Select the correct Answer for each question.
Use a triple integral to find the volume of the solid bounded by and the planes and

A)
B)
C)
D)
E)

Question 70

Select the correct Answer for each question.
Use spherical coordinates.Evaluate where is the ball with center the origin and radius

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

Question 71

Select the correct Answer for each question.
Calculate the iterated integral.

A)
B)4
C)
D)
E)None of these

Question 72

Select the correct Answer for each question.
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)9.447
B)3.296
C)8.841
D)4.447
E)5.088

Question 73

Select the correct Answer for each question.
Find the area of the part of hyperbolic paraboloid that lies between the cylinders

A)
B)
C)
D)
E)

Question 74

Select the correct Answer for each question.
Find the volume of the solid bounded by the surface and the planes and coordinate planes.

A)
B)
C)
D)
E)

Question 75

Select the correct Answer for each question.
Find the mass and the center of mass of the lamina occupying the region R, where R is the triangular region with vertices and having the mass density

A)
B)
C)
D)

Question 76

Select the correct Answer for each question.
Evaluate the iterated integral.

A)
B)
C)
D)
E)

Question 77

Select the correct Answer for each question.
Use cylindrical coordinates to evaluate where T is the solid bounded by the cylinder and the planes

A)
B)
C)
D)

Question 78

Select the correct Answer for each question.
Evaluate the integral and with respect to in that order.

A)28
B)36
C)4
D)24

Question 79

Select the correct Answer for each question.
Use polar coordinates to find the volume of the solid inside the cylinder and the ellipsoid

A)
B)
C)
D)
E)

Question 80

Select the correct Answer for each question.
Use cylindrical coordinates to evaluate the triple integral where E is the solid that lies between the cylinders above the xy-plane and below the plane

A)8.57
B)0
C)3.4
D)9.19
E)0.54