Deck 6: Applications of the Integral

Question 1

Find the area of the shaded region bounded by the graphs of and and the axis.

Answer 1

Question 2

Find the volume of the solid shown in the figure if the radius of the upper circle is , the radius of the bottom circle is and the height of the solid is .

Answer 2

Question 3

Find the area of the region enclosed by the curves and .

Answer 3

Question 4

The base of a solid is the region bounded by the -axis and the semi-ellipse Each cross section perpendicular to the base and parallel to the -axis is an isosceles triangle whose height is twice its base. Find the volume of this solid.

Question 5

Find the area of the shaded region below which is bounded by the graphs of and .

Question 6

Calculate the area of the region enclosed by and for .

Question 7

Find the area of the region shown in the figure. Use symmetry to facilitate your computation.

Question 8

Find the area of the region bounded by the graphs of and and the y-axis.

Question 9

Find the area of the region bounded by and for .

Question 10

Find the mass of a rod of length 2 with density function , where is the distance from one of the rod's ends.

Question 11

Find the mass of a semicircular disc of radius for which the density function is , where is the distance from the diameter.

Question 12

Find the area enclosed by the graphs and .

Question 13

Let . Find an equation for in the interval such that is equal to the average of on . Use Newton's Method with . Give your answer to three decimal places.

Question 14

Calculate the total area between the graphs of the functions and .

Question 15

Find the area of the region bounded by the graphs of and .

Question 16

Find the area of the shaded region below which is bounded by the graphs of and .

Question 17

Find the area of the region bounded by the graphs of the functions and .

Question 18

Find the area of the shaded region shown in the figure.

Question 19

Find the volume of the solid whose base is the semi-circular region and the cross sections perpendicular to the y-axis are isosceles right triangles whose hypotenuse lie on the base of the solid.

Question 20

Find the area of the region bounded by the graphs of and .

Question 21

Compute the volume of the solid whose base is the region between the x-axis and the curve over , and the cross sections parallel to the y-axis and perpendicular to the base are semicircles.

Question 22

Find the mass of the solid whose base is the region inside the unit circle , and the cross sections parallel to the x-axis and perpendicular to the base are semicircles whose diameters are on the base.
The density function is .

Question 23

Calculate the population within a 2-mile radius of a city center if the radial population density is thousands per square mile.

Question 24

Find the volume of the solid obtained by revolving the region about the line .

Question 25

Calculate the population within a 2-mile radius of a city center if the radial population density is thousands per square mile.

Question 26

A solid circular cylinder of radius and length whose density is the square of the distance from the axis of symmetry has the following mass:

A) .
B) .
C) .
D) .
E) .

Question 27

Compute the volume of the solid whose base is the region between the x-axis and the curve over , and the cross sections parallel to the y-axis and perpendicular to the base are squares.

Question 28

The volume of the solid obtained by rotating the ellipse about the y-axis is

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

Question 29

Find the value of in such that is equal to the average of on where .

Question 30

Find the mass of a rod of length aligned on the positive -axis if its mass density equals the area of the right triangle shown in the figure.

Question 31

Find a value in such that is equal to the average of on if

Question 32

Find the volume of the solid obtained by revolving the region about the x-axis.

Question 33

Find the volume of the solid obtained by rotating the region about the line .

Question 34

Find the volume generated by revolving the triangle shown in the figure about side .

Question 35

Find the volume generated by revolving a right triangle with angle and hypotenuse 2 about a line perpendicular to the hypotenuse passing through the vertex of the angle, as shown in the figure.

Question 36

The volume of the ice goblet obtained by rotating the region shown in the figure about the y-axis is

A) .
B) .
C) .
D) .
E) .

Question 37

Calculate the population within a 4-mile radius of a city center if the radial population density is thousands per square mile.

Question 38

What is the average volume of a regular tetrahedron with side if varies from 1 to 4?

Question 39

Find the volume of the solid obtained by revolving the region in the first quadrant between the curves and about the line .

Question 40

Find the value of in such that is equal to the average of on .

Question 41

Find the volume of the solid obtained by rotating the region shown in the figure about the line .

Question 42

The volume of the solid obtained by rotating the region shown in the figure about the y-axis is

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

Question 43

Find the volume of the solid obtained by rotating the region about the line .

Question 44

Find the volume of the solid obtained by rotating the region between the curve and the x-axis above the interval about the line .

Question 45

Find the volume of the solid obtained by rotating the region between the curves and about the line . Use the formula .

Question 46

Find the volume of the solid obtained by rotating the region in the first quadrant enclosed by the curves and for , about the line .

Question 47

Use the Shell Method to find the volume of the solid obtained by rotating the region shown in the figure about the y-axis.

Question 48

Use the Shell Method to calculate the volume of the solid obtained by rotating the region shown in the figure about the line .

Question 49

Find the volume generated by revolving an isosceles right triangle with hypotenuse 2 about a line perpendicular to the hypotenuse passing through the vertex of one of the acute angles.

Question 50

The volume of the solid obtained by revolving the region between the lines and about the y-axis is

A) .
B) .
C) .
D) .
E) .

Question 51

Find the volume of the solid generated by revolving the region below over about the line .

Question 52

Use the Shell Method to calculate the volume of the solid obtained by rotating the region shown in the figure about the line .

Question 53

Use the Shell Method to calculate the volume of the solid obtained by rotating the region about the line .

Question 54

To compute the volume of the solid obtained by rotating the region shown in the figure about the y-axis, the most convenient method is

A) the Disk Method.
B) the Disk Method in addition to the Shell Method.
C) the Shell Method.
D) Integration by Parts.
E) none of these methods.

Question 55

Find the volume generated by revolving the triangle shown in the figure about a line parallel to the hypotenuse, passing through the vertex of the right angle.

Question 56

To compute the volume of the solid obtained by rotating the region between the line and the curve in the first quadrant about the y-axis, the most convenient method is

A) the Shell Method.
B) the Disk Method.
C) the Shell Method in addition to the Disk Method.
D) Integration by Parts.
E) none of these methods.

Question 57

Calculate the volume of the solid obtained by rotating the region between the curves and over about the line .

Question 58

Use the Shell Method to calculate the volume of the solid obtained by rotating the region shown in the figure about the y-axis.

Question 59

Use the Shell Method to calculate the volume of the solid obtained by rotating the region shown in the figure about the y-axis.

Question 60

Use the Shell Method to calculate the volume of the solid obtained by rotating the sector shown in the figure about the line Use the formula .

Question 61

Find the approximate work required to pump water out of the container shown in the figure if the container is filled with water to the top of the cylinder. The density of water is .

Question 62

Find the work required to build a pyramid of height , whose base is an isosceles right triangle with a leg length of . The pyramid will be built with stones of density .

A)
B)
C)
D)
E)

Question 63

To compute the volume of the solid obtained by rotating the region in the figure about the y-axis, the most convenient method is

A) the Disk Method.
B) the Shell Method.
C) both the Disk and Shell Methods.
D) Integration by Parts.
E) none of these methods.

Question 64

Find the work required to move an object along the x-axis from to by the force .

Question 65

A conical container (vertex down) is filled to half of its height with liquid of density . The radius of the conical container is and its height is , measured in meters.
The approximate work required to pump the liquid out of the top of the tank is

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

Question 66

The vertical tank shown in the figure is half full of water. Given that the density of water is , the approximate work required to empty the tank by pumping the water out through an outlet at the top of the tank is

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

Question 67

A vertical tank, with elliptic horizontal cross section and height is full of water. The work done in pumping the water out through an outlet at the top of the tank is which of the following? (Assume that the water weighs 62.5 pounds per cubic foot.)

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

Question 68

Calculate the volume of the solid obtained by rotating the region shown in the figure about the line .

Question 69

Find the work done in moving an object from to by the force .

Question 70

The volume of the solid obtained by rotating the region shown in the figure about the
Y-axis is

A) .
B) .
C) .
D) .
E) .

Question 71

Use the Shell Method to calculate the volume of the solid obtained by rotating the region shown in the figure about the line .

Question 72

The work (against gravity) required to build a right pyramid of height , with equilateral triangle base of side length , using stones of density is

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

Question 73

If Joules of work are needed to stretch a spring cm beyond equilibrium, the spring constant is

A) .
B) .
C) .
D) .
E) .

Question 74

A vertical cylindrical tank of radius 5 feet and height 12 feet is full of water. The water weighs 62.5 pounds per cubic foot. Find the work done in pumping out the water:

A) through an outlet at the top of the tank.
B) through an outlet 6 feet above the top of the tank.

Question 75

A chain weighing 25 pounds per foot is hanging from the top of a 100-foot building, as shown in the figure. Find the work needed to pull the chain to the top of the building.

Question 76

Calculate the volume of the solid obtained by rotating the region in the figure about the y-axis. (Note: the x and y scales are different.)

Question 77

Find the work done in moving an object from to by the force .

Question 78

Find the work done in moving an object from to by the force .

Question 79

To compute the volume of the solid obtained by rotating the region shown in the figure about the y-axis, the most convenient method is

A) the Disk Method.
B) the Shell Method.
C) the Disk and Shell Methods.
D) Integration by Parts.
E) none of these methods.

Question 80

Use the Shell Method to find the volume of the solid obtained by rotating the region shown in the figure about the line .