Question 1

Find the area of the region bounded by the given curves. Write a single integral that represents the area.

Accepted Answer

A)  area   
B)  area   
C)  area   
D)  area   
A)  area   
B)  area   
C)  area   
D)  area

Question 2

Identify the mass of a thin wire with density   .

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 3

Use the best method available to find the volume of the solid formed by revolving the region bounded by     and   about the y-axis. Write the integral that is used to find the volume.

Accepted Answer

The answer of Use the best method available to find...

Question 4

In a baseball game a base-runner can attempt to advance when a fielder catches a hit ball before it reaches the ground, so long as the runner doesn't leave base before the fielder catches the ball. The fielder then attempts to throw the ball to the base ahead of the runner. If a base-runner can run from 3rd base to home plate in 4.6 sec, how fast must the centerfielder's throw leave his hand if he makes the catch then immediately throws the ball 370 ft from home plate? Assume the base-runner leaves his base at the same instant the centerfielder throws the ball, and that the ball reaches the catcher standing over home plate at the same height from which it was thrown and at the same time the runner would reach the plate. [Hint: This is basically a problem of determining how much initial vertical velocity the ball needs to stay in the air long enough to reach the plate.]

Accepted Answer

A)  80 ft/sec 
B)  74 ft/sec 
C)  109 ft/sec 
D)  154 ft/sec 
A)  80 ft/sec 
B)  74 ft/sec 
C)  109 ft/sec 
D)  154 ft/sec

Question 5

Compute the weight in ounces of an object extending from x = -5 to x = 25 with density   slugs/in. Round answer to nearest whole number.

Accepted Answer

A)  21 oz 
B)  4 oz 
C)  86 oz 
D)  443 oz 
A)  21 oz 
B)  4 oz 
C)  86 oz 
D)  443 oz

Question 6

Find the volume of the solid with cross-sectional area   extending over the range   .

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 7

Compute the arc length exactly.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 8

The moment of inertia, I, of an object is the second moment of its mass distribution relative to an axis of rotation,   , where r is the perpendicular distance from the rotation axis and   is the density of the object at a distance r from the rotation axis. What is the moment of inertia for rotation about the y-axis of a shape described by the region bordered by   x = 0, and y = 0 revolved about the y-axis? Assume the density of the material is 1.0, so that mass and volume will be numerically equivalent. [Hint: Cylindrical shells are particularly conducive to moment-of-inertia calculations since r is everywhere the same for a given shell.]

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 9

An anthill is in the shape formed by revolving the region bounded by   and the x-axis about the y-axis. A researcher removes a cylindrical core from the center of the hill. What should the radius be to give the researcher 5% of the dirt?

Accepted Answer

A)  0.8409 
B)  0.6818 
C)  0.1591 
D)  0.3182 
A)  0.8409 
B)  0.6818 
C)  0.1591 
D)  0.3182

Question 10

Pick the integral that represents a volume calculation by cylindrical shells that is equivalent to the following integral that represents a calculation of the same volume but by the method of washers.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 11

Identify the graph and the area bounded by the curves   on the interval   .

Accepted Answer

A)      
B)      
C)      
D)      
A)      
B)      
C)      
D)

Question 12

Find the area between the curves on the given interval. y = x4, y = x - 1, -1  $\leq$ x $\leq$ 1

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 13

A solid is formed by revolving the given region about the given line. Round to three decimal places. Region bounded by y = sec x, y = 0, x =   , and x =   about y = -6

Accepted Answer

A)  72.737 
B)  23.153 
C)  250.39 
D)  19.348 
A)  72.737 
B)  23.153 
C)  250.39 
D)  19.348

Question 14

A 30-foot chain weighs 750 pounds and is hauled up to the deck of a boat. The chain is oriented vertically and the top of the chain starts 30 feet below the deck. Compute the work done.

Accepted Answer

A)  11,250 foot-pounds 
B)  22,875 foot-pounds 
C)  375 foot-pounds 
D)  33,750 foot-pounds 
A)  11,250 foot-pounds 
B)  22,875 foot-pounds 
C)  375 foot-pounds 
D)  33,750 foot-pounds

Question 15

Where is the center of mass of a region of uniform density bounded by   ?

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 16

Starting with the expression for work   , one can change the look of the expression without changing what it represents by dividing the force F by the area it is applied over and multiplying the differential distance dx by that same area. Force divided by area is a pressure (P), and area times the differential distance is a differential volume. Hence one can also describe work as   . A gas at constant temperature will change its pressure inversely to changes in volume,   . If a sample of gas has a pressure of 1 atmosphere when its volume is 1 L, how much work does it do when it expands from 1 L to 3 L?

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 17

Identify the graph and the area of the region determined by the intersections of the tabulated curves using Simpson's Rule. x
0)00
0)25
0)50
0)75
1)00
F1(x)
1)0
0)25
0)07
0)02
0)00
F2(x)
1)0
0)75
0)50
0)25
0)00

Accepted Answer

A)      
B)      
C)      
D)      
A)      
B)      
C)      
D)

Question 18

Use Simpson's Rule to estimate the volume of the shape obtained by revolving the cross-section given in the table about the y-axis. y
0
0)4375
0)75
0)9375
1
0)9375
0)75
0)4375
0
X
1
1)25
1)5
1)75
2
2)25
2)5
2)75
3

Accepted Answer

A)  3 
B)  17 
C)  24 
D)  26 
A)  3 
B)  17 
C)  24 
D)  26

Question 19

Sketch the solid bounded by   and the x-axis on the interval   revolved about the line   Draw a typical shell and write an integral that can be used to compute the volume of the solid.

Accepted Answer

The answer of Sketch the solid bounded by  ...

Question 20

Find the volume of the solid formed by revolving the region bounded by   about x = 5.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Find the area of the region bounded by the given curves. Write a single integral that represents the area.

Identify the mass of a thin wire with density .

Use the best method available to find the volume of the solid formed by revolving the region bounded by and about the y-axis. Write the integral that is used to find the volume.

Compute the weight in ounces of an object extending from x = -5 to x = 25 with density slugs/in. Round answer to nearest whole number.

Find the volume of the solid with cross-sectional area extending over the range .

Compute the arc length exactly.

An anthill is in the shape formed by revolving the region bounded by and the x-axis about the y-axis. A researcher removes a cylindrical core from the center of the hill. What should the radius be to give the researcher 5% of the dirt?

Pick the integral that represents a volume calculation by cylindrical shells that is equivalent to the following integral that represents a calculation of the same volume but by the method of washers.

Identify the graph and the area bounded by the curves on the interval .

Find the area between the curves on the given interval. y = x⁴, y = x - 1, -1 $\leq$ x $\leq$ 1

A solid is formed by revolving the given region about the given line. Round to three decimal places. Region bounded by y = sec x, y = 0, x = , and x = about y = -6

A 30-foot chain weighs 750 pounds and is hauled up to the deck of a boat. The chain is oriented vertically and the top of the chain starts 30 feet below the deck. Compute the work done.

Where is the center of mass of a region of uniform density bounded by ?

Identify the graph and the area of the region determined by the intersections of the tabulated curves using Simpson's Rule. x 0)00 0)25 0)50 0)75 1)00 F₁(x) 1)0 0)25 0)07 0)02 0)00 F₂(x) 1)0 0)75 0)50 0)25 0)00

Use Simpson's Rule to estimate the volume of the shape obtained by revolving the cross-section given in the table about the y-axis. y 0 0)4375 0)75 0)9375 1 0)9375 0)75 0)4375 0 X 1 1)25 1)5 1)75 2 2)25 2)5 2)75 3

Sketch the solid bounded by and the x-axis on the interval revolved about the line Draw a typical shell and write an integral that can be used to compute the volume of the solid.

Find the volume of the solid formed by revolving the region bounded by about x = 5.

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Exponentials, Logarithms and Other Transcendental Functions

Integration Techniques

First-Order Differential Equations

Infinite Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables and Partial Differentiation

Multiple Integrals

Vector Calculus

Second Order Differential Equations

Filters

Exam 6: Applications of the Definite Integral

Find the area of the region bounded by the given curves. Write a single integral that represents the area.

Identify the mass of a thin wire with density .

Use the best method available to find the volume of the solid formed by revolving the region bounded by and about the y-axis. Write the integral that is used to find the volume.

Compute the weight in ounces of an object extending from x = -5 to x = 25 with density slugs/in. Round answer to nearest whole number.

Find the volume of the solid with cross-sectional area extending over the range .

Compute the arc length exactly.

An anthill is in the shape formed by revolving the region bounded by and the x-axis about the y-axis. A researcher removes a cylindrical core from the center of the hill. What should the radius be to give the researcher 5% of the dirt?

Pick the integral that represents a volume calculation by cylindrical shells that is equivalent to the following integral that represents a calculation of the same volume but by the method of washers.

Identify the graph and the area bounded by the curves on the interval .

Find the area between the curves on the given interval. y = x4, y = x - 1, -1 ≤\leq≤ x ≤\leq≤ 1

A solid is formed by revolving the given region about the given line. Round to three decimal places. Region bounded by y = sec x, y = 0, x = , and x = about y = -6

A 30-foot chain weighs 750 pounds and is hauled up to the deck of a boat. The chain is oriented vertically and the top of the chain starts 30 feet below the deck. Compute the work done.

Where is the center of mass of a region of uniform density bounded by ?

Identify the graph and the area of the region determined by the intersections of the tabulated curves using Simpson's Rule. x 0)00 0)25 0)50 0)75 1)00 F1(x) 1)0 0)25 0)07 0)02 0)00 F2(x) 1)0 0)75 0)50 0)25 0)00

Use Simpson's Rule to estimate the volume of the shape obtained by revolving the cross-section given in the table about the y-axis. y 0 0)4375 0)75 0)9375 1 0)9375 0)75 0)4375 0 X 1 1)25 1)5 1)75 2 2)25 2)5 2)75 3

Sketch the solid bounded by and the x-axis on the interval revolved about the line Draw a typical shell and write an integral that can be used to compute the volume of the solid.

Find the volume of the solid formed by revolving the region bounded by about x = 5.

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Exponentials, Logarithms and Other Transcendental Functions

Integration Techniques

First-Order Differential Equations

Infinite Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables and Partial Differentiation

Multiple Integrals

Vector Calculus

Second Order Differential Equations

Filters

Find the area between the curves on the given interval. y = x⁴, y = x - 1, -1 $\leq$ x $\leq$ 1

Identify the graph and the area of the region determined by the intersections of the tabulated curves using Simpson's Rule. x 0)00 0)25 0)50 0)75 1)00 F₁(x) 1)0 0)25 0)07 0)02 0)00 F₂(x) 1)0 0)75 0)50 0)25 0)00