Find the center of mass of the point masses lying on the x-axis. $\begin{array} { l } m _ { 1 } = 7 , m _ { 2 } = 10 , m _ { 3 } = 8 , m _ { 4 } = 10 , m _ { 5 } = 8 \\ x _ { 1 } = - 9 , x _ { 2 } = - 2 , x _ { 3 } = - 6 , x _ { 4 } = 9 , x _ { 5 } = 6 \end{array}$

A) $\bar { x } = \frac { 9 } { 43 }$ B) $\bar { x } = \frac { 10 } { 43 }$ C) $\bar { x } = \frac { 5 } { 23 }$ D) $\bar { x } = \frac { 6 } { 43 }$ E) $\bar { x } = \frac { 7 } { 43 }$ A) $\bar { x } = \frac { 9 } { 43 }$ B) $\bar { x } = \frac { 10 } { 43 }$ C) $\bar { x } = \frac { 5 } { 23 }$ D) $\bar { x } = \frac { 6 } { 43 }$ E) $\bar { x } = \frac { 7 } { 43 }$

Find the area of the region bounded by the graphs of the algebraic functions. \[\begin{array} { l } f ( x ) = x ^ { 2 } + 30 x + 225 \\ g ( x ) = 17 ( x + 15 ) \end{array}\]

A) $A = \frac { 4913 } { 12 }$ B) $A = \frac { 5363 } { 6 }$ C) $A = \frac { 4913 } { 6 }$ D) $A = \frac { 6263 } { 6 }$ E) $A = \frac { 4913 } { 3 }$ A) $A = \frac { 4913 } { 12 }$ B) $A = \frac { 5363 } { 6 }$ C) $A = \frac { 4913 } { 6 }$ D) $A = \frac { 6263 } { 6 }$ E) $A = \frac { 4913 } { 3 }$

Find the center of mass of the point masses lying on the x-axis. $\begin{array} { l } m _ { 1 } = 10 , m _ { 2 } = 1 , m _ { 3 } = 6 , m _ { 4 } = 5 \\ x _ { 1 } = 2 , x _ { 2 } = - 10 , x _ { 3 } = - 7 , x _ { 4 } = - 8 \end{array}$

A) $\bar { x } = - \frac { 73 } { 22 }$ B) $\bar { x } = - \frac { 69 } { 22 }$ C) $\bar { x } = - \frac { 69 } { 25 }$ D) $\bar { x } = - \frac { 35 } { 11 }$ E) $\bar { x } = - \frac { 36 } { 11 }$ A) $\bar { x } = - \frac { 73 } { 22 }$ B) $\bar { x } = - \frac { 69 } { 22 }$ C) $\bar { x } = - \frac { 69 } { 25 }$ D) $\bar { x } = - \frac { 35 } { 11 }$ E) $\bar { x } = - \frac { 36 } { 11 }$

Find the center of mass of the given system of point masses. $\begin{array}{|c|c|c|c|c|c|} \hline m_{i} & 8 & 8 & 8 & 7 & 9 \\ \hline\left(x_{i}, y_{i}\right) & (5,-2) & (6,-7) & (2,4) & (-7,-7) & (-5,5) \\ \hline \end{array}$

A) $\bar { x } = \frac { 13 } { 43 } , \bar { y } = - \frac { 41 } { 43 }$ B) $\bar { x } = \frac { 9 } { 40 } , \bar { y } = - \frac { 9 } { 8 }$ C) $\bar { x } = \frac { 1 } { 4 } , \bar { y } = - \frac { 11 } { 10 }$ D) $\bar { x } = \frac { 13 } { 40 } , \bar { y } = - \frac { 41 } { 40 }$ E) $\bar { x } = \frac { 3 } { 10 } , \bar { y } = - \frac { 21 } { 20 }$ A) $\bar { x } = \frac { 13 } { 43 } , \bar { y } = - \frac { 41 } { 43 }$ B) $\bar { x } = \frac { 9 } { 40 } , \bar { y } = - \frac { 9 } { 8 }$ C) $\bar { x } = \frac { 1 } { 4 } , \bar { y } = - \frac { 11 } { 10 }$ D) $\bar { x } = \frac { 13 } { 40 } , \bar { y } = - \frac { 41 } { 40 }$ E) $\bar { x } = \frac { 3 } { 10 } , \bar { y } = - \frac { 21 } { 20 }$

Exam 7: Area of a Region Between Two Curves

Use the shell method to find the volume of the solid generated by revolving the plane region bounded by $y = 4 x ^ { 2 } , y = 10 x - x ^ { 2 }$ , about the line $x = 2$

(Multiple Choice)

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Question 61

$\text { The vertical cross section of an irrigation canal is modeled by } f ( x ) = \frac { 5 x ^ { 2 } } { \left( x ^ { 2 } + 4 \right) } \text {, }$ where $x$ is measured in feet and $x = 0$ corresponds to the center of the canal. Use the integration capabilities of a graphing utility to approximate the fluid force against a vertical gate used to stop the flow of water if the water is 3 feet deep. Round your answer to three decimal places.

(Multiple Choice)

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Question 62

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. $y = x ^ { 2 } , y = 8 x - x ^ { 2 }$ (i) $x$ -axis; (ii) the line $y = 18$

(Multiple Choice)

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Question 63

Set up and evaluate the integral that gives the volume of the solid formed by revolving the region bounded by $y = x ^ { \frac { 3 } { 4 } } , y = 1$ , and $x = 0$ about the $y$ -axis.

(Multiple Choice)

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Question 64

A force of 20 pounds stretches a spring 11 inches in an exercise machine. Find the work done in stretching the spring 2 feet from its natural position.

(Multiple Choice)

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Question 65

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 half full, assuming that the diameter is Feet and the gasoline weighs 42 pounds per cubic foot.

(Multiple Choice)

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Question 66

Two electrons repel each other with a force that varies inversely as the square of the distance between them, where $k$ is the constant of proportionality. One electron is fixed at the point $( 2,2 )$ . Find the work done in moving the second electron from $( - 4,2 )$ to $( 1,2 )$ .

(Multiple Choice)

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Question 67

Find the center of mass of the point masses lying on the x-axis. =7,=10,=8,=10,=8 =-9,=-2,=-6,=9,=6

(Multiple Choice)

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Question 68

A tank on the wing of a jet aircraft is formed by revolving the region bounded by the graph of $y = \frac { 1 } { 15 } x ^ { 2 } \sqrt { 2 - x }$ and the $x$ -axis $( 0 \leq x \leq 2 )$ about the $x$ -axis, where $x$ and $y$ are measured in meters. Find the volume of the tank. Round your answer to two decimal places.

(Multiple Choice)

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Question 69

$\text { Find } M _ { y } \text { for the lamina of uniform density } \rho \text { bounded by the graphs of the }$ equations $x = 4 y + 12$ and $x = y ^ { 2 }$ .

(Multiple Choice)

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Question 70

Use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations $y = 2 x ^ { 3 } , y = 0 , x = 3$ about the $x$ -axis.

(Multiple Choice)

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Question 71

Find the area of the region bounded by the graphs of the algebraic functions. f(x)=+30x+225 g(x)=17(x+15)

(Multiple Choice)

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Question 72

If the accumulation function $F ( x ) \text { is given by } F ( x ) = \int _ { 0 } ^ { x } \left( \frac { 1 } { 11 } t ^ { 2 } + 5 \right) d t \text {, evaluate }$ $F( 9 )$

(Multiple Choice)

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Question 73

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line $y = 8$ $y = x , y = 7 , x = 0$

(Multiple Choice)

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Question 74

Consider a beam of length $L = 5 \text { feet with a fulcrum } x \text { feet from one end as shown in }$ the figure. In order to move a 550-pound object, a person weighing 214 pounds wants to balance it on the beam. Find x (the distance between the person and the fulcrum) such that the system is Equilibrium. Round your answer to two decimal places. $Consider a beam of length L = 5 \text { feet with a fulcrum } x \text { feet from one end as shown in } the figure. In order to move a 550-pound object, a person weighing 214 pounds wants to balance it on the beam. Find x (the distance between the person and the fulcrum) such that the system is Equilibrium. Round your answer to two decimal places.$

(Multiple Choice)

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Question 75

$\text { Find the arc length of the graph of the function } y = \frac { 2 } { 3 } x ^ { \frac { 3 } { 2 } } + 2 \text { over the interval [14, }$ $16$ ].

(Multiple Choice)

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Question 76

ِAn overhead garage door has two springs, one on each side of the door. A force of 12 pounds is required to stretch each spring 1 foot. Because of the pulley system, the springs stretch only One-half the distance the door travels. The door moves a total of 10 feet and springs are at their Natural length when the door is open. Find the work done by pair of strings.

(Multiple Choice)

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Question 77

Concrete sections for the new building have the dimensions (in meters) and shape as shown in the figure (the picture is not necessarily drawn to scale). Find the area of the face of the Section superimposed on the rectangular coordinate system. Round your answer to three decimal Places.

(Multiple Choice)

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Question 78

Find the center of mass of the point masses lying on the x-axis. =10,=1,=6,=5 =2,=-10,=-7,=-8

(Multiple Choice)

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Question 79

Find the center of mass of the given system of point masses. 8 8 8 7 9 , (5,-2) (6,-7) (2,4) (-7,-7) (-5,5)

(Multiple Choice)

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Question 80

Use the shell method to find the volume of the solid generated by revolving the plane region bounded by $y = 4 x ^ { 2 } , y = 10 x - x ^ { 2 }$ , about the line $x = 2$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. $y = x ^ { 2 } , y = 8 x - x ^ { 2 }$ (i) $x$ -axis; (ii) the line $y = 18$

Set up and evaluate the integral that gives the volume of the solid formed by revolving the region bounded by $y = x ^ { \frac { 3 } { 4 } } , y = 1$ , and $x = 0$ about the $y$ -axis.

A force of 20 pounds stretches a spring 11 inches in an exercise machine. Find the work done in stretching the spring 2 feet from its natural position.

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 half full, assuming that the diameter is Feet and the gasoline weighs 42 pounds per cubic foot.

Two electrons repel each other with a force that varies inversely as the square of the distance between them, where $k$ is the constant of proportionality. One electron is fixed at the point $( 2,2 )$ . Find the work done in moving the second electron from $( - 4,2 )$ to $( 1,2 )$ .

Find the center of mass of the point masses lying on the x-axis. =7,=10,=8,=10,=8 =-9,=-2,=-6,=9,=6

$\text { Find } M _ { y } \text { for the lamina of uniform density } \rho \text { bounded by the graphs of the }$ equations $x = 4 y + 12$ and $x = y ^ { 2 }$ .

Use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations $y = 2 x ^ { 3 } , y = 0 , x = 3$ about the $x$ -axis.

Find the area of the region bounded by the graphs of the algebraic functions. f(x)=+30x+225 g(x)=17(x+15)

If the accumulation function $F ( x ) \text { is given by } F ( x ) = \int _ { 0 } ^ { x } \left( \frac { 1 } { 11 } t ^ { 2 } + 5 \right) d t \text {, evaluate }$ $F( 9 )$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line $y = 8$ $y = x , y = 7 , x = 0$

$\text { Find the arc length of the graph of the function } y = \frac { 2 } { 3 } x ^ { \frac { 3 } { 2 } } + 2 \text { over the interval [14, }$ $16$ ].

Concrete sections for the new building have the dimensions (in meters) and shape as shown in the figure (the picture is not necessarily drawn to scale). Find the area of the face of the Section superimposed on the rectangular coordinate system. Round your answer to three decimal Places.

Find the center of mass of the point masses lying on the x-axis. =10,=1,=6,=5 =2,=-10,=-7,=-8

Find the center of mass of the given system of point masses. 8 8 8 7 9 , (5,-2) (6,-7) (2,4) (-7,-7) (-5,5)

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 shell method to find the volume of the solid generated by revolving the plane region bounded by y=4x2,y=10x−x2y = 4 x ^ { 2 } , y = 10 x - x ^ { 2 }y=4x2,y=10x−x2 , about the line x=2x = 2x=2

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. y=x2,y=8x−x2y = x ^ { 2 } , y = 8 x - x ^ { 2 }y=x2,y=8x−x2 (i) xxx -axis; (ii) the line y=18y = 18y=18

Set up and evaluate the integral that gives the volume of the solid formed by revolving the region bounded by y=x34,y=1y = x ^ { \frac { 3 } { 4 } } , y = 1y=x43​,y=1 , and x=0x = 0x=0 about the yyy -axis.

A force of 20 pounds stretches a spring 11 inches in an exercise machine. Find the work done in stretching the spring 2 feet from its natural position.

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 half full, assuming that the diameter is Feet and the gasoline weighs 42 pounds per cubic foot.

Find the center of mass of the point masses lying on the x-axis. =7,=10,=8,=10,=8 =-9,=-2,=-6,=9,=6

Use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations y=2x3,y=0,x=3y = 2 x ^ { 3 } , y = 0 , x = 3y=2x3,y=0,x=3 about the xxx -axis.

Find the area of the region bounded by the graphs of the algebraic functions. f(x)=+30x+225 g(x)=17(x+15)

If the accumulation function F(x) is given by F(x)=∫0x(111t2+5)dt, evaluate F ( x ) \text { is given by } F ( x ) = \int _ { 0 } ^ { x } \left( \frac { 1 } { 11 } t ^ { 2 } + 5 \right) d t \text {, evaluate }F(x) is given by F(x)=∫0x​(111​t2+5)dt, evaluate F(9)F( 9 )F(9)

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y=8y = 8y=8 y=x,y=7,x=0y = x , y = 7 , x = 0y=x,y=7,x=0

Concrete sections for the new building have the dimensions (in meters) and shape as shown in the figure (the picture is not necessarily drawn to scale). Find the area of the face of the Section superimposed on the rectangular coordinate system. Round your answer to three decimal Places.

Find the center of mass of the point masses lying on the x-axis. =10,=1,=6,=5 =2,=-10,=-7,=-8

Find the center of mass of the given system of point masses. 8 8 8 7 9 , (5,-2) (6,-7) (2,4) (-7,-7) (-5,5)

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 shell method to find the volume of the solid generated by revolving the plane region bounded by $y = 4 x ^ { 2 } , y = 10 x - x ^ { 2 }$ , about the line $x = 2$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. $y = x ^ { 2 } , y = 8 x - x ^ { 2 }$ (i) $x$ -axis; (ii) the line $y = 18$

Set up and evaluate the integral that gives the volume of the solid formed by revolving the region bounded by $y = x ^ { \frac { 3 } { 4 } } , y = 1$ , and $x = 0$ about the $y$ -axis.

Use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations $y = 2 x ^ { 3 } , y = 0 , x = 3$ about the $x$ -axis.

If the accumulation function $F ( x ) \text { is given by } F ( x ) = \int _ { 0 } ^ { x } \left( \frac { 1 } { 11 } t ^ { 2 } + 5 \right) d t \text {, evaluate }$ $F( 9 )$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line $y = 8$ $y = x , y = 7 , x = 0$