Deck 9: Further Applications of the Integral and Taylor Polynomials

Question 1

Approximate the arc length of the curve over the interval using Simpson's Rule .

Answer 1

4.27

Question 2

Let be the solid obtained by revolving the infinite graph of about the x-axis for . Which of the following statements is correct?

A) has infinite volume and finite surface area.
B) has finite volume and infinite surface area.
C) has finite volume and finite surface area.
D) has infinite volume and infinite surface area.
E) None of the above is correct.

Answer 2

has finite volume and infinite surface area.

Question 3

Approximate the arc length of the curve over the interval using the Trapezoidal Rule .

Answer 3

21.956

Question 4

Compute the surface area of revolution of about the x-axis over the interval .

Question 5

Calculate the arc length of over the interval .

Question 6

Calculate the arc length of the curve over the interval .

Question 7

Compute the area of the surface obtained by rotating the graph of , about the x-axis.

Question 8

Compute the surface area of revolution of about the x-axis over the interval .

Question 9

Compute the surface area of revolution of about the x-axis over the interval .

Question 10

Calculate the arc length of the curve over the interval .

Question 11

Compute the arc length of for .

Question 12

Compute the surface area of revolution of about the x-axis over the interval .

Question 13

Use implicit differentiation to compute the surface area of revolution about the x-axis of the part of the astroid in the first quadrant.

Question 14

Find the area of the surface obtained by rotating the graph of , about the x-axis.

Question 15

Calculate the arc length of over the interval [1, 5].

Question 16

Let be the solid obtained by revolving the infinite graph of about the x-axis for . Which of the following statements is correct?

A) has finite volume and finite surface area.
B) has finite surface area and infinite volume.
C) has finite volume and infinite surface area.
D) has infinite volume and infinite surface area.
E) None of the above is correct.

Question 17

Compute the surface area of revolution of about the x-axis over the interval .

Question 18

Compute the arc length of over the interval [0, 2].

Question 19

Find the area of the surface obtained by rotating the loop , , about the x-axis.
Use implicit differentiation to facilitate your computations.

Question 20

Calculate the arc length of over the interval .

Question 21

The plate shown in the figure, determined by the region enclosed by the graphs of , , and , is submerged vertically in a fluid with density so that its top edge is level with the surface of the fluid. Find the force on a side of the plate if is given in terms of weight per unit volume.

Question 22

The face of a dam is an isosceles trapezoid of height 30 ft and bases 10 ft and 50 ft. The dam is oriented vertically with the larger base at the bottom. Find the force on the dam if the water level is 4 ft below the top of the dam. (The density of the water is .)

Question 23

The plate shown in the figure is submerged in water so that its top is level with the surface of the water. Find the fluid pressure on a side of the plate. (The density of water is .)

Question 24

An infinite plate shown in the figure below, bounded by the graphs of and and the y-axis, is submerged vertically in water, with its top ft below the water surface.
Calculate the fluid force on a side of the plate. (The density of water is )

Question 25

The plate shown in the figure, enclosed by the curves , , and , is submerged vertically in a fluid with density so that its top is at a depth of . Find the fluid pressure on a side of the plate if is given in terms of weight per unit volume.

Question 26

An infinite plate shown in the figure below, bounded by the graphs of and and the y-axis, is submerged vertically in water, with its top ft below the water surface.
Calculate the fluid force on a side of the plate. (The density of water is )

Question 27

A thin plate shown in the figure below, bounded by the curves , , and is submerged vertically in water, with its top level with the water's surface. Calculate the fluid force on a side of the plate.
(The density of water is )

Question 28

Calculate the arc length of over the interval

Question 29

A water tank is a cylinder 4 ft in diameter, standing on its base. If water fills the tank to a depth of 3 ft, what is the magnitude of the force exerted on the side of the tank? (The density of the water is .)

Question 30

An infinite plate shown in the figure below, bounded by the graphs of and and the two axes, is submerged vertically in a fluid with density in kilograms per cubic meter; its top is level with the fluid surface.
Calculate the fluid force on a side of the plate and write the answer in terms of and .

Question 31

The plate shown in the figure below, bounded by the graph of and the
y-axis, is submerged vertically in a fluid with density so that its top is level with the fluid surface.
Find the fluid pressure on a side of the plate if is given in terms of weight per unit volume.

Question 32

The plate determined by the region between the graphs of and is submerged in a fluid with density , with its top touching the surface of the fluid.
Find the fluid pressure on a side of the plate if is given in terms of weight per unit volume.

Question 33

Compute the arc length of for .

Question 34

A thin plate shown in the figure, bounded by the graphs and and the x-axis, is submerged vertically in a fluid of density so that its top is level with the fluid surface.
Calculate the fluid pressure on a side of the plate if is given in terms of weight per unit volume.

Question 35

An infinite plate shown in the figure below, bounded by the graphs of , , and the y-axis, is submerged vertically in water, with its top ft below the water surface. Calculate the fluid force on a side of the plate.
(The density of water is )

Question 36

Approximate the arc length of the curve over the interval using Simpson's Rule .

Question 37

Find the surface area of the ellipsoid obtained by rotating the ellipse about the x-axis.

Question 38

A thin plate shown in the following figure is bounded by the graphs of and . The plate is submerged in a fluid with density so that its top is level with the surface of the fluid.
Calculate the fluid pressure on a side of the plate if is given in terms of weight per unit volume.

Question 39

Let be the region shown in the figure below enclosed by the graph of ,
the positive x-axis, and the negative y-axis.
Calculate the fluid force on a side of the plate in the shape if the water surface is at .
(The density of water is w = )

Question 40

A thin plate shown in the figure below, bounded by the curves and , is submerged vertically in water with its top ft below the water surface. Calculate the fluid force on a side of the plate.
(The density of water is )

Question 41

A right triangle is submerged vertically in water so that the base is the part of the triangle which is closest to the surface. Suppose that the base of the triangle is 2 ft long, and the height is 6 ft. What is the fluid force on a side of the triangle if the base is 3 ft from the surface? (The density of water is )

Question 42

Find the center of mass of the region enclosed by the graphs of and .

Question 43

Find the centroid of the shaded region in the following figure.

Question 44

An infinite plate occupying the region bounded by the functions and on the interval is submerged vertically so that the top of the plate is under 1.5 m of water. What is the fluid force on the side of the plate?

Question 45

A rectangular box with height 2 m, width 3 m, and length 4 m is submerged in a pool of water. The top of the box is 5 m below the surface of the water.
Compute the fluid force on one of the box sides with dimensions 4 m by 2 m.

Question 46

Find the centroid of the quarter ring shown in the figure below.

Question 47

Find the centroid of the shaded region in the figure below.

Question 48

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

Question 49

A plate bounded by the functions and on the interval is submerged vertically so that the top of the plate is under 1 m of water. What is the fluid force on the side of the plate?

Question 50

A plate occupying the region bounded by the functions and is submerged vertically so that the top of the plate is under 2 m of water. What is the fluid force on the side of the plate?

Question 51

A plate bounded by the functions , , and is submerged vertically so that the top of the plate is under 4 m of water. What is the fluid force on the side of the plate?

Question 52

Use additivity of moments to find the center of mass of the region consisting of a semicircle on top of an isosceles trapezoid of height 1 and bases 2 and 4, as shown below.

Question 53

Find the centroid of the region in the figure below enclosed by the circle and the two tangents to the circle intersecting at the point .

Question 54

Find the centroid of the region lying between the graphs of and over the interval .

Question 55

The centroid of the region enclosed by the graphs of and lies on the line

A)
B)
C)
D)
E)

Question 56

Let be an invertible function such that and . Let be the region enclosed by the graphs of and over the interval . The centroid of lies on

A) the line .
B) the line .
C) the line .
D) It depends on the function .
E) none of the above.

Question 57

A water tank is a rectangular tank with a square base of side length 4 ft and a height of 3 ft, standing on its base. If oil fills the tank to a depth of 2 ft, what is the magnitude of the force exerted on each of the vertical sides of the tank? (The density of the oil is .)

Question 58

A plate occupying the region bounded by the functions and is submerged vertically so that the top of the plate is under 2 m of water. What is the fluid force on the side of the plate?

Question 59

A plate occupying the region bounded by the functions and is submerged vertically so that the top of the plate is under 5 m of water. What is the fluid force on the side of the plate?

Question 60

Find the centroid of the shaded region in the figure below.

Question 61

Find the center of mass of the region enclosed by the graphs of and .

Question 62

Calculate the Taylor polynomial centered at for .

Question 63

Find the centroid of the region enclosed by the graphs , , , and

Question 64

The quotient is equal to:

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

Question 65

Let be the Taylor polynomial of centered at . Which of the following statements is correct?

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

Question 66

Compute for centered at . Use the error bound to find the maximum possible size of error of .

Question 67

Compute for centered at . Use the error bound to find the maximum possible size of error of .

Question 68

Which of the following is a Maclaurin polynomial of ?

A)
B)
C)
D)
E)

Question 69

Find the centroid of the portion of the unit circle lying within the first three quadrants.

Question 70

Find the centroid of the region enclosed by the graph of , the tangent to the graph at and the y-axis.

Question 71

Five particles of equal mass are located at and . Find the center of mass of the system.

Question 72

Find the centroid of the shaded region in the figure below.

Question 73

In the following figure, the centroid of the region enclosed by the graphs of , and lies on:

A) the line
B) the line
C) the line
D) the graph of
E) the graph of

Question 74

Find for the lamina of uniform density occupying the region under from .

Question 75

Find the centroid of the shaded region shown in the figure below.

Question 76

Calculate the Maclaurin polynomial for .

Question 77

Which of the following are Maclaurin polynomials of

A)
B)
C)
D)
E)

Question 78

The centroid of the region shown in the figure below enclosed by the graphs of , , and lies on:

A) the line
B) the line
C) the line
D) the graph of
E) the graph of

Question 79

Four particles with masses of 4, 5, 7, and 3 are located respectively at and . Find such that the center of mass of the system is located at the point .

Question 80

Calculate the Maclaurin polynomial for .