Question 1

$\text { Find the arc length of the graph of the function } y = 8 x ^ { \frac { 3 } { 2 } } + 7 \text { over the interval } [ 0,6 ] \text {. }$

Accepted Answer

A)  $\frac { 865 \sqrt { 865 } - 1 } { 216 }$ 
B)  $\frac { 865 \sqrt { 865 } - 864 } { 216 }$ 
C)  $\frac { 865 \sqrt { 865 } + 1 } { 432 }$ 
D)  $\frac { 865 \sqrt { 865 } + 1 } { 216 }$ 
E)  $\frac { 865 \sqrt { 865 } - 1 } { 432 }$ 
A)  $\frac { 865 \sqrt { 865 } - 1 } { 216 }$ 
B)  $\frac { 865 \sqrt { 865 } - 864 } { 216 }$ 
C)  $\frac { 865 \sqrt { 865 } + 1 } { 432 }$ 
D)  $\frac { 865 \sqrt { 865 } + 1 } { 216 }$ 
E)  $\frac { 865 \sqrt { 865 } - 1 } { 432 }$

Question 2

A solid is generated by revolving the region bounded by $y = \frac { 1 } { 12 } x ^ { 2 } \text { and } y = 12$ about the y-axis. A hole, centered along the axis of revolution, is drilled through this solid so that one-fourth of the volume is removed. Find the diameter of the hole. Round your answer
To three decimal places.

Accepted Answer

A) 10.733 
B) 11.167 
C) 4.392 
D) 2.536 
E) 8.785 
A) 10.733 
B) 11.167 
C) 4.392 
D) 2.536 
E) 8.785

Question 3

Use the Theorem of Pappus to find the volume of the solid formed by revolving the region bounded by the graphs of $y = x , y = 10$, and $x = 0$ about the $x$-axis. Round your answer to two decimal places.

Accepted Answer

A)  $1809.56$ 
B)  $2787.64$ 
C)  $2094.40$ 
D)  $3141.59$ 
E)  $3619.11$ 
A)  $1809.56$ 
B)  $2787.64$ 
C)  $2094.40$ 
D)  $3141.59$ 
E)  $3619.11$

Question 4

surface of a machine part is the region between the graphs of $y _ { 1 } = | x | \text { and }$ $y _ { 2 } = 0.100 x ^ { 2 } + 2.500$ as shown in the figure. Find the area of the surface of the machine part. Round your answer to five decimal places.

Accepted Answer

A)  $1.66667$ 
B)  $0.41667$ 
C)  $0.01333$ 
D)  $33.33333$ 
E)  $8.33333$ 
A)  $1.66667$ 
B)  $0.41667$ 
C)  $0.01333$ 
D)  $33.33333$ 
E)  $8.33333$

Question 5

Find the center of mass of the given system of point masses. $\begin{array}{|c|c|c|c|}
\hline m_{i} & 9 & 4 & 7 \
\hline\left(x_{i}, y_{i}\right) & (-1,0) & (7,-3) & (1,-9) \
\hline
\end{array}$

Accepted Answer

A)  $\bar { x } = \frac { 5 } { 4 } , \bar { y } = - \frac { 19 } { 5 }$ 
B)  $\bar { x } = \frac { 13 } { 10 } , \bar { y } = - \frac { 15 } { 4 }$ 
C)  $\bar { x } = \frac { 29 } { 23 } , \bar { y } = - \frac { 72 } { 23 }$ 
D)  $\bar { x } = \frac { 29 } { 20 } , \bar { y } = - \frac { 18 } { 5 }$ 
E)  $\bar { x } = \frac { 7 } { 5 } , \bar { y } = - \frac { 73 } { 20 }$ 
A)  $\bar { x } = \frac { 5 } { 4 } , \bar { y } = - \frac { 19 } { 5 }$ 
B)  $\bar { x } = \frac { 13 } { 10 } , \bar { y } = - \frac { 15 } { 4 }$ 
C)  $\bar { x } = \frac { 29 } { 23 } , \bar { y } = - \frac { 72 } { 23 }$ 
D)  $\bar { x } = \frac { 29 } { 20 } , \bar { y } = - \frac { 18 } { 5 }$ 
E)  $\bar { x } = \frac { 7 } { 5 } , \bar { y } = - \frac { 73 } { 20 }$

Question 6

Determine the work done by lifting a 140 pound bag of sugar 12 feet.

Accepted Answer

A) 1,680 ft-lb 
B) 16,800 ft-lb 
C) 2,600 ft-lb 
D) 168 ft-lb 
E) 260 ft-lb 
A) 1,680 ft-lb 
B) 16,800 ft-lb 
C) 2,600 ft-lb 
D) 168 ft-lb 
E) 260 ft-lb

Question 7

A cylindrical gasoline tank is placed so that the axis of the cylinder is horizontal. Find the fluid force on a circular end of the tank if the tank is full, assuming that the diameter is Feet and the gasoline weighs pounds per cubic foot. (Evaluate one integral by a geometric formula And the other by observing that the integrand is an odd function.) Round your answer to two decimal
Places.

Accepted Answer

A)  $135.09 \mathrm { lb }$ 
B)  $128.81 \mathrm { lb }$ 
C)  $64.40 \mathrm { lb }$ 
D)  $1,030.44 \mathrm { lb }$ 
E)  $122.52 \mathrm { lb }$ 
A)  $135.09 \mathrm { lb }$ 
B)  $128.81 \mathrm { lb }$ 
C)  $64.40 \mathrm { lb }$ 
D)  $1,030.44 \mathrm { lb }$ 
E)  $122.52 \mathrm { lb }$

Question 8

$\text { Find } ( \bar { x } , \bar { y } ) \text { for the lamina of uniform density } \rho \text { bounded by the graphs of the }$ equations $y = \sqrt { x } , y = 0 , x = 4$.

Accepted Answer

A)  $( \bar { x } , \bar { y } ) = \left( \frac { 12 } { 5 } , \frac { 3 } { 4 } \right)$ 
B)  $( \bar { x } , \bar { y } ) = \left( 0 , \frac { 3 } { 4 } \right)$ 
C)  $( \bar { x } , \bar { y } ) = \left( \frac { 6 } { 5 } , \frac { 3 } { 4 } \right)$ 
D)  $( \bar { x } , \bar { y } ) = \left( \frac { 3 } { 4 } , \frac { 12 } { 5 } \right)$ 
E)  $( \bar { x } , \bar { y } ) = \left( \frac { 12 } { 5 } , 0 \right)$ $\text { Find } M _ { x } \text { for the lamina of uniform density } \rho \text { bounded by the graphs of the }$ 
A)  $( \bar { x } , \bar { y } ) = \left( \frac { 12 } { 5 } , \frac { 3 } { 4 } \right)$ 
B)  $( \bar { x } , \bar { y } ) = \left( 0 , \frac { 3 } { 4 } \right)$ 
C)  $( \bar { x } , \bar { y } ) = \left( \frac { 6 } { 5 } , \frac { 3 } { 4 } \right)$ 
D)  $( \bar { x } , \bar { y } ) = \left( \frac { 3 } { 4 } , \frac { 12 } { 5 } \right)$ 
E)  $( \bar { x } , \bar { y } ) = \left( \frac { 12 } { 5 } , 0 \right)$ $\text { Find } M _ { x } \text { for the lamina of uniform density } \rho \text { bounded by the graphs of the }$

Question 9

A bridge has a main span of about 1400 meters. Each of its towers has a height of about 146 meters. Approximate the length of a parabolic cable along the main span if the length C of $\text { the cable is given by } C = 2 \int _ { 0 } ^ { w } \sqrt { 1 + \left( 4 h ^ { 2 } / w ^ { 4 } \right) x ^ { 2 } } d x \text {. Round your answer to one decimal place. }$

Accepted Answer

A) 719.8 meters 
B) 759.4 meters 
C) 2120.7 meters 
D) 1439.6 meters 
E) 2879.2 meters 
A) 719.8 meters 
B) 759.4 meters 
C) 2120.7 meters 
D) 1439.6 meters 
E) 2879.2 meters

Question 10

Set up and evaluate the definite integral for the area of the surface formed by revolving the graph of $y = \frac { x } { 2 } , 1 \leq x \leq 9$ about the $x$-axis.

Accepted Answer

A)  $40 \sqrt { 5 } \pi$ 
B)  $4 \sqrt { 5 } \pi$ 
C)  $20 \sqrt { 5 }$ 
D)  $4 \sqrt { 5 }$ 
E)  $20 \sqrt { 5 } \pi$ 
A)  $40 \sqrt { 5 } \pi$ 
B)  $4 \sqrt { 5 } \pi$ 
C)  $20 \sqrt { 5 }$ 
D)  $4 \sqrt { 5 }$ 
E)  $20 \sqrt { 5 } \pi$

Question 11

$\text { Find the arc length of the graph of the function } y = \frac { x ^ { 3 } } { 6 } + \frac { 1 } { 2 x } \text { over the interval } [ 1,2 ] \text {. }$

Accepted Answer

A)  $\frac { 15 } { 4 }$ 
B)  $\frac { 49 } { 12 }$ 
C)  $\frac { 17 } { 12 }$ 
D)  $\frac { 15 } { 8 }$ 
E)  $\frac { 49 } { 24 }$ 
A)  $\frac { 15 } { 4 }$ 
B)  $\frac { 49 } { 12 }$ 
C)  $\frac { 17 } { 12 }$ 
D)  $\frac { 15 } { 8 }$ 
E)  $\frac { 49 } { 24 }$

Question 12

Neglecting air resistance and the weight of the propellant, determine the work done in propelling a 12-ton satellite to a height of 100 miles above Earth. Assume that the Earth has a Radius of 4000 miles.

Accepted Answer

A) 1,756.10 mi-ton 
B) 1,170.73 mi-ton 
C) 702.44 mi-ton 
D) 936.59 mi-ton 
E) 2,575.61 mi-ton 
A) 1,756.10 mi-ton 
B) 1,170.73 mi-ton 
C) 702.44 mi-ton 
D) 936.59 mi-ton 
E) 2,575.61 mi-ton

Question 13

Find the area of the region bounded by the graphs of the function $f ( x ) = \frac { 9 x } { x ^ { 2 } + 1 } , y = 0,0 \leq x \leq 3$. Round your answer to three decimal places.

Accepted Answer

A)  $20.723$ 
B)  $11.182$ 
C)  $6.238$ 
D)  $10.362$ 
E)  $22.364$ 
A)  $20.723$ 
B)  $11.182$ 
C)  $6.238$ 
D)  $10.362$ 
E)  $22.364$

Question 14

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations $y = 2 x ^ { 2 } , y = 0$, and $x = 2$ about the line $x = 2$.

Accepted Answer

A)  $\frac { 16 } { 3 } \pi$ 
B)  $\frac { 8 } { 3 } \pi$ 
C)  $\frac { 16 } { 3 }$ 
D)  $\frac { 32 } { 3 }$ 
E)  $\frac { 32 } { 3 } \pi$ 
A)  $\frac { 16 } { 3 } \pi$ 
B)  $\frac { 8 } { 3 } \pi$ 
C)  $\frac { 16 } { 3 }$ 
D)  $\frac { 32 } { 3 }$ 
E)  $\frac { 32 } { 3 } \pi$

Question 15

A force of 265 Newtons stretches a spring 20 centimeters. How much work is done in stretching the spring from 15 centimeters to 25 centimeters?

Accepted Answer

A) 28.5 joules 
B) 31.5 joules 
C) 26.5 joules 
D) 21.5 joules 
E) 36.5 joules 
A) 28.5 joules 
B) 31.5 joules 
C) 26.5 joules 
D) 21.5 joules 
E) 36.5 joules

Question 16

Set up the definite integral that gives the area of the region bounded by the graph of $y _ { 1 } = x ^ { 2 } + 2 x + 1$ and $y _ { 2 } = 2 x + 5$

Accepted Answer

A)  $\int _ { - 2 } ^ { 2 } \left( - x ^ { 2 } + 4 \right) d x$ 
B)  $\int _ { - 2 } ^ { 2 } \left( x ^ { 2 } + 2 x + 1 \right) d x$ 
C) $\int _ { - 2 } ^ { 2 } \left( - x ^ { 2 } + 4 \right) d y$ 
D)  $\int _ { - 2 } ^ { 2 } \left( x ^ { 2 } + 4 x + 6 \right) d y$ 
E)  $\int _ { - 2 } ^ { 2 } \left( x ^ { 2 } + 4 x + 6 \right) d x$ 
A)  $\int _ { - 2 } ^ { 2 } \left( - x ^ { 2 } + 4 \right) d x$ 
B)  $\int _ { - 2 } ^ { 2 } \left( x ^ { 2 } + 2 x + 1 \right) d x$ 
C) $\int _ { - 2 } ^ { 2 } \left( - x ^ { 2 } + 4 \right) d y$ 
D)  $\int _ { - 2 } ^ { 2 } \left( x ^ { 2 } + 4 x + 6 \right) d y$ 
E)  $\int _ { - 2 } ^ { 2 } \left( x ^ { 2 } + 4 x + 6 \right) d x$

Question 17

Find the fluid force of a triangular vertical plate submerged in water, where the dimensions are given in meters $( a = 15 , b = 6 , c = 4 )$ and the weight-density of water is 9800 newtons per cubic meter.

Accepted Answer

A)  $6,174,000$ newtons 
B)  $4,851,000$ newtons 
C)  $8,379,000$ newtons 
D)  $4,483,500$ newtons 
E)  $14,994,000$ newtons 
A)  $6,174,000$ newtons 
B)  $4,851,000$ newtons 
C)  $8,379,000$ newtons 
D)  $4,483,500$ newtons 
E)  $14,994,000$ newtons

Question 18

Use the integration capabilities of a graphing utility to approximate the arc length of the graph of $y = \cos x \text { on the interval } [ 0 , \pi ]$ . Round your answer to three decimal places.

Accepted Answer

A) 3.620 
B) 5.020 
C) 10.540 
D) 3.820 
E) 5.270 
A) 3.620 
B) 5.020 
C) 10.540 
D) 3.820 
E) 5.270

Question 19

$\text { Find the volume of the solid generated by rotating the circle } ( x - 10 ) ^ { 2 } + y ^ { 2 } = 16$ about the y-axis.

Accepted Answer

A)  $V = 320 \pi ^ { 2 }$ 
B)  $V = 80 \pi ^ { 2 }$ 
C)  $V = 80 \pi$ 
D)  $V = 320 \pi$ 
E)  $V = 160 \pi ^ { 2 }$ 
A)  $V = 320 \pi ^ { 2 }$ 
B)  $V = 80 \pi ^ { 2 }$ 
C)  $V = 80 \pi$ 
D)  $V = 320 \pi$ 
E)  $V = 160 \pi ^ { 2 }$

Question 20

$\text { Find the arc length of the graph of the function } x = \frac { 1 } { 3 } \left( y ^ { 2 } + 2 \right) ^ { \frac { 3 } { 2 } } \text { over the interval }$ $0 \leq y \leq 11$

Accepted Answer

A)  $\frac { 1363 } { 3 }$ 
B)  $\frac { 1364 } { 3 }$ 
C)  $\frac { 1361 } { 6 }$ 
D)  $\frac { 1363 } { 6 }$ 
E)  $\frac { 455 } { 2 }$ 
A)  $\frac { 1363 } { 3 }$ 
B)  $\frac { 1364 } { 3 }$ 
C)  $\frac { 1361 } { 6 }$ 
D)  $\frac { 1363 } { 6 }$ 
E)  $\frac { 455 } { 2 }$

$\text { Find the arc length of the graph of the function } y = 8 x ^ { \frac { 3 } { 2 } } + 7 \text { over the interval } [ 0,6 ] \text {. }$

Use the Theorem of Pappus to find the volume of the solid formed by revolving the region bounded by the graphs of $y = x , y = 10$ , and $x = 0$ about the $x$ -axis. Round your answer to two decimal places.

Find the center of mass of the given system of point masses. 9 4 7 , (-1,0) (7,-3) (1,-9)

Determine the work done by lifting a 140 pound bag of sugar 12 feet.

$\text { Find } ( \bar { x } , \bar { y } ) \text { for the lamina of uniform density } \rho \text { bounded by the graphs of the }$ equations $y = \sqrt { x } , y = 0 , x = 4$ .

Set up and evaluate the definite integral for the area of the surface formed by revolving the graph of $y = \frac { x } { 2 } , 1 \leq x \leq 9$ about the $x$ -axis.

$\text { Find the arc length of the graph of the function } y = \frac { x ^ { 3 } } { 6 } + \frac { 1 } { 2 x } \text { over the interval } [ 1,2 ] \text {. }$

Neglecting air resistance and the weight of the propellant, determine the work done in propelling a 12-ton satellite to a height of 100 miles above Earth. Assume that the Earth has a Radius of 4000 miles.

Find the area of the region bounded by the graphs of the function $f ( x ) = \frac { 9 x } { x ^ { 2 } + 1 } , y = 0,0 \leq x \leq 3$ . Round your answer to three decimal places.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations $y = 2 x ^ { 2 } , y = 0$ , and $x = 2$ about the line $x = 2$ .

A force of 265 Newtons stretches a spring 20 centimeters. How much work is done in stretching the spring from 15 centimeters to 25 centimeters?

Set up the definite integral that gives the area of the region bounded by the graph of $y _ { 1 } = x ^ { 2 } + 2 x + 1$ and $y _ { 2 } = 2 x + 5$ $Set up the definite integral that gives the area of the region bounded by the graph of y _ { 1 } = x ^ { 2 } + 2 x + 1 and y _ { 2 } = 2 x + 5$

Find the fluid force of a triangular vertical plate submerged in water, where the dimensions are given in meters $( a = 15 , b = 6 , c = 4 )$ and the weight-density of water is 9800 newtons per cubic meter.

Use the integration capabilities of a graphing utility to approximate the arc length of the graph of $y = \cos x \text { on the interval } [ 0 , \pi ]$ . Round your answer to three decimal places.

$\text { Find the volume of the solid generated by rotating the circle } ( x - 10 ) ^ { 2 } + y ^ { 2 } = 16$ about the y-axis.

$\text { Find the arc length of the graph of the function } x = \frac { 1 } { 3 } \left( y ^ { 2 } + 2 \right) ^ { \frac { 3 } { 2 } } \text { over the interval }$ $0 \leq y \leq 11$

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Exam 7: Area of a Region Between Two Curves

Use the Theorem of Pappus to find the volume of the solid formed by revolving the region bounded by the graphs of y=x,y=10y = x , y = 10y=x,y=10 , and x=0x = 0x=0 about the xxx -axis. Round your answer to two decimal places.

Find the center of mass of the given system of point masses. 9 4 7 , (-1,0) (7,-3) (1,-9)

Determine the work done by lifting a 140 pound bag of sugar 12 feet.

Set up and evaluate the definite integral for the area of the surface formed by revolving the graph of y=x2,1≤x≤9y = \frac { x } { 2 } , 1 \leq x \leq 9y=2x​,1≤x≤9 about the xxx -axis.

Neglecting air resistance and the weight of the propellant, determine the work done in propelling a 12-ton satellite to a height of 100 miles above Earth. Assume that the Earth has a Radius of 4000 miles.

Find the area of the region bounded by the graphs of the function f(x)=9xx2+1,y=0,0≤x≤3f ( x ) = \frac { 9 x } { x ^ { 2 } + 1 } , y = 0,0 \leq x \leq 3f(x)=x2+19x​,y=0,0≤x≤3 . Round your answer to three decimal places.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations y=2x2,y=0y = 2 x ^ { 2 } , y = 0y=2x2,y=0 , and x=2x = 2x=2 about the line x=2x = 2x=2 .

A force of 265 Newtons stretches a spring 20 centimeters. How much work is done in stretching the spring from 15 centimeters to 25 centimeters?

Set up the definite integral that gives the area of the region bounded by the graph of y1=x2+2x+1y _ { 1 } = x ^ { 2 } + 2 x + 1y1​=x2+2x+1 and y2=2x+5y _ { 2 } = 2 x + 5y2​=2x+5

Find the fluid force of a triangular vertical plate submerged in water, where the dimensions are given in meters (a=15,b=6,c=4)( a = 15 , b = 6 , c = 4 )(a=15,b=6,c=4) and the weight-density of water is 9800 newtons per cubic meter.

Use the integration capabilities of a graphing utility to approximate the arc length of the graph of y=cos⁡x on the interval [0,π]y = \cos x \text { on the interval } [ 0 , \pi ]y=cosx on the interval [0,π] . Round your answer to three decimal places.

Find the volume of the solid generated by rotating the circle (x−10)2+y2=16\text { Find the volume of the solid generated by rotating the circle } ( x - 10 ) ^ { 2 } + y ^ { 2 } = 16 Find the volume of the solid generated by rotating the circle (x−10)2+y2=16 about the y-axis.

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Use the Theorem of Pappus to find the volume of the solid formed by revolving the region bounded by the graphs of $y = x , y = 10$ , and $x = 0$ about the $x$ -axis. Round your answer to two decimal places.

Set up and evaluate the definite integral for the area of the surface formed by revolving the graph of $y = \frac { x } { 2 } , 1 \leq x \leq 9$ about the $x$ -axis.

Find the area of the region bounded by the graphs of the function $f ( x ) = \frac { 9 x } { x ^ { 2 } + 1 } , y = 0,0 \leq x \leq 3$ . Round your answer to three decimal places.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations $y = 2 x ^ { 2 } , y = 0$ , and $x = 2$ about the line $x = 2$ .

Set up the definite integral that gives the area of the region bounded by the graph of $y _ { 1 } = x ^ { 2 } + 2 x + 1$ and $y _ { 2 } = 2 x + 5$ $Set up the definite integral that gives the area of the region bounded by the graph of y _ { 1 } = x ^ { 2 } + 2 x + 1 and y _ { 2 } = 2 x + 5$

Find the fluid force of a triangular vertical plate submerged in water, where the dimensions are given in meters $( a = 15 , b = 6 , c = 4 )$ and the weight-density of water is 9800 newtons per cubic meter.

Use the integration capabilities of a graphing utility to approximate the arc length of the graph of $y = \cos x \text { on the interval } [ 0 , \pi ]$ . Round your answer to three decimal places.

$\text { Find the volume of the solid generated by rotating the circle } ( x - 10 ) ^ { 2 } + y ^ { 2 } = 16$ about the y-axis.