Question 1

Let V be the volume of the solid generated by revolving the region enclosed by $y = \sqrt { x },$ the x-axis, x = 4; about $y = 3$ . Then V is

Accepted Answer

A)  $\sqrt { 3 } \pi$ 
B)  $8 \pi$ 
C)  $24 \pi$ 
D)  $\frac { 8 \pi } { 3 }$ 
E)  $\frac { 8 \pi } { 5 }$ 
A)  $\sqrt { 3 } \pi$ 
B)  $8 \pi$ 
C)  $24 \pi$ 
D)  $\frac { 8 \pi } { 3 }$ 
E)  $\frac { 8 \pi } { 5 }$ C

Question 2

Let V be the volume of the solid generated by revolving the region enclosed by y = sec x, the x-axis, $x = \frac { \pi } { 3 };$ about the x-axis. Then V is

Accepted Answer

A)  $\sqrt { 3 } \pi$ 
B)  $\frac { 128 \pi } { 3 }$ 
C)  $\frac { 32 \pi } { 5 }$ 
D)  $\frac { 8 \pi } { 3 }$ 
E)  $\frac { 8 \pi } { 5 }$ 
A)  $\sqrt { 3 } \pi$ 
B)  $\frac { 128 \pi } { 3 }$ 
C)  $\frac { 32 \pi } { 5 }$ 
D)  $\frac { 8 \pi } { 3 }$ 
E)  $\frac { 8 \pi } { 5 }$ A

Question 3

A cylindrical sewer pipe of radius $\frac { 1 } { 2 }$ foot is half full of water of density 62.5 pounds per cubic foot. A gate used to seal off the sewer is placed perpendicular to the pipe opening. The force, in pounds, due to hydrostatic pressure on one end of the gate is

Accepted Answer

A)  $\frac { 250 } { 3 }$ 
B)  $\frac { 500 } { 3 }$ 
C)  $1,125$ 
D)  $\frac { 1,000 } { 3 }$ 
E)  $\frac { 125 } { 24 }$ 
A)  $\frac { 250 } { 3 }$ 
B)  $\frac { 500 } { 3 }$ 
C)  $1,125$ 
D)  $\frac { 1,000 } { 3 }$ 
E)  $\frac { 125 } { 24 }$ E

Question 4

Three particles having masses 1, 2, and 3 kg are located on the x-axis at -1,2,3 respectively. The center of mass of this system is located at x =

Accepted Answer

A)  $\frac { 1,125 } { 8 }$ 
B)  $\frac { 500 } { 3 }$ 
C) 2 
D)  $\frac { 15,625 } { 24 }$ 
E)  $\frac { 125 } { 24 }$ 
A)  $\frac { 1,125 } { 8 }$ 
B)  $\frac { 500 } { 3 }$ 
C) 2 
D)  $\frac { 15,625 } { 24 }$ 
E)  $\frac { 125 } { 24 }$

Question 5

Which of these integrals can be used to calculate the surface area of the solid formed by revolving the region bounded by $f ( x ) = x ^ { 4 } - 3 x,$ the x-axis, x = -2, and x = 0 about the x-axis?

Accepted Answer

A)  $2 \pi \int _ { - 2 } ^ { 0 } \left( x ^ { 4 } - 3 x \right) \sqrt { 1 + \left( 4 x ^ { 3 } - 3 \right) ^ { 2 } } d x$ 
B)  $2 \pi \int _ { - 2 } ^ { 0 } \left( 4 x ^ { 3 } - 3 \right) \sqrt { 1 + \left( x ^ { 4 } - 3 x \right) ^ { 2 } } d x$ 
C)  $\int _ { - 2 } ^ { 0 } \left( x ^ { 4 } - 3 x \right) \sqrt { 1 + \left( 4 x ^ { 3 } - 3 \right) } d x$ 
D)  $\int _ { 0 } ^ { - 2 } \left( 4 x ^ { 3 } - 3 \right) \sqrt { 1 + \left( x ^ { 4 } - 3 x \right) ^ { 2 } } d x$ 
E)  $2 \pi \int _ { 0 } ^ { - 2 } \left( x ^ { 4 } - 3 x \right) \sqrt { 1 + \left( 4 x ^ { 3 } - 3 \right) ^ { 2 } } d x$ 
A)  $2 \pi \int _ { - 2 } ^ { 0 } \left( x ^ { 4 } - 3 x \right) \sqrt { 1 + \left( 4 x ^ { 3 } - 3 \right) ^ { 2 } } d x$ 
B)  $2 \pi \int _ { - 2 } ^ { 0 } \left( 4 x ^ { 3 } - 3 \right) \sqrt { 1 + \left( x ^ { 4 } - 3 x \right) ^ { 2 } } d x$ 
C)  $\int _ { - 2 } ^ { 0 } \left( x ^ { 4 } - 3 x \right) \sqrt { 1 + \left( 4 x ^ { 3 } - 3 \right) } d x$ 
D)  $\int _ { 0 } ^ { - 2 } \left( 4 x ^ { 3 } - 3 \right) \sqrt { 1 + \left( x ^ { 4 } - 3 x \right) ^ { 2 } } d x$ 
E)  $2 \pi \int _ { 0 } ^ { - 2 } \left( x ^ { 4 } - 3 x \right) \sqrt { 1 + \left( 4 x ^ { 3 } - 3 \right) ^ { 2 } } d x$

Question 6

Let V be the volume of the solid generated by revolving the region enclosed by about the x-axis. Then V is

Accepted Answer

A)  $\frac { \pi } { 3 }$ 
B)  $\frac { 704 \pi } { 5 }$ 
C)  $\frac { \pi ( 4 - \pi ) } { 2 }$ 
D)  $\frac { 8 \pi } { 3 }$ 
E)  $\pi$ 
A)  $\frac { \pi } { 3 }$ 
B)  $\frac { 704 \pi } { 5 }$ 
C)  $\frac { \pi ( 4 - \pi ) } { 2 }$ 
D)  $\frac { 8 \pi } { 3 }$ 
E)  $\pi$

Question 7

By the Pappus Theorem, the volume of the solid formed by revolving the region enclosed by the circle $( x - 3 ) ^ { 2 } + y ^ { 2 } = 1$ about the y-axis is

Accepted Answer

A) 1 
B)  $\frac { 2 } { 5 }$ 
C)  $\frac { 6 } { 5 }$ 
D)  $\frac { 3 } { 4 }$ 
E)  $6 \pi ^ { 2 }$ 
A) 1 
B)  $\frac { 2 } { 5 }$ 
C)  $\frac { 6 } { 5 }$ 
D)  $\frac { 3 } { 4 }$ 
E)  $6 \pi ^ { 2 }$

Question 8

The arc length of $y = \frac { x ^ { \frac { 3 } { 2 } } } { 3 } - x ^ { \frac { 1 } { 2 } } \text { for } x \in [ 1,4 ]$ is

Accepted Answer

A)  $\frac { 16 } { 5 }$ 
B)  $\frac { 10 } { 3 }$ 
C)  $\frac { 23 } { 8 }$ 
D)  $\frac { 55 } { 3 }$ 
E)  $\frac { 87 } { 8 }$ 
A)  $\frac { 16 } { 5 }$ 
B)  $\frac { 10 } { 3 }$ 
C)  $\frac { 23 } { 8 }$ 
D)  $\frac { 55 } { 3 }$ 
E)  $\frac { 87 } { 8 }$

Question 9

Let $\bar { x }$ denote the x-coordinate of the centroid of the region enclosed by $y = x ^ { 2 }$ and $y = 1$ Then $\bar { x }$ =

Accepted Answer

A) 1 
B)  $\frac { 2 } { 5 }$ 
C)  $\frac { 93 } { 35 }$ 
D)  $\frac { 3 } { 4 }$ 
E) 0 
A) 1 
B)  $\frac { 2 } { 5 }$ 
C)  $\frac { 93 } { 35 }$ 
D)  $\frac { 3 } { 4 }$ 
E) 0

Question 10

A cylindrical sewer pipe of radius $\frac { 1 } { 2 }$ feet is half full of water of density 62.5 pounds per cubic foot. A gate used to seal off the sewer is placed perpendicular to the pipe opening. The force, in pounds, due to hydrostatic pressure on one end of the gate is

Accepted Answer

A)  $\frac { 1,125 } { 8 }$ 
B)  $\frac { 500 } { 3 }$ 
C)  $1,125$ 
D)  $\frac { 1,000 } { 3 }$ 
E)  $\frac { 125 } { 24 }$ 
A)  $\frac { 1,125 } { 8 }$ 
B)  $\frac { 500 } { 3 }$ 
C)  $1,125$ 
D)  $\frac { 1,000 } { 3 }$ 
E)  $\frac { 125 } { 24 }$

Question 11

Let A denote the area enclosed by the equations $y ^ { 2 } = x$ and $y = 2 - x$ Then A is

Accepted Answer

A)  $\frac { 11 } { 2 }$ 
B)  $\frac { 9 } { 2 }$ 
C)  $\frac { 7 } { 2 }$ 
D)  $\frac { 5 } { 2 }$ 
E)  $\frac { 3 } { 2 }$ 
A)  $\frac { 11 } { 2 }$ 
B)  $\frac { 9 } { 2 }$ 
C)  $\frac { 7 } { 2 }$ 
D)  $\frac { 5 } { 2 }$ 
E)  $\frac { 3 } { 2 }$

Question 12

The arc length of $y = x ^ { \frac { 3 } { 2 } } \text { for } x \in [ 1,8 ]$ is

Accepted Answer

A)  $\frac{152 . \sqrt{19} - 12 \sqrt{13}}{27}$ 
B)  $\frac { 13 \sqrt { 13 } - 152 \sqrt { 19 }} { 27 }$ 
C)  $\frac { 152 \sqrt { 19 } + 12\sqrt { 13 } } { 27 }$ 
D)  $\frac { 152 \sqrt { 19 } + 13\sqrt { 13 } } { 27 }$ 
E)  $\frac { 152 \sqrt { 19 } - 12\sqrt { 13 } } { 27 }$ 
A)  $\frac{152 . \sqrt{19} - 12 \sqrt{13}}{27}$ 
B)  $\frac { 13 \sqrt { 13 } - 152 \sqrt { 19 }} { 27 }$ 
C)  $\frac { 152 \sqrt { 19 } + 12\sqrt { 13 } } { 27 }$ 
D)  $\frac { 152 \sqrt { 19 } + 13\sqrt { 13 } } { 27 }$ 
E)  $\frac { 152 \sqrt { 19 } - 12\sqrt { 13 } } { 27 }$

Question 13

The arc length of $y = - ( x - 1 ) ^ { \frac { 3 } { 2 } } - 1 \text { for } x \in [ 5,10 ]$ is

Accepted Answer

A)  $\frac { 5 ( 17 \sqrt { 85 } + 16 \sqrt { 10 } ) } { 27 }$ 
B)  $\frac { 5 ( 17 \sqrt { 85 } - 16 \sqrt { 10 } ) } { 27 }$ 
C)  $\frac { 5 ( 85 \sqrt { 17 } - 16 \sqrt { 10 } ) } { 27 }$ 
D)  $\frac { 5 ( 85 \sqrt { 17 } + 16 \sqrt { 10 } ) } { 27 }$ 
E)  $\frac { 5 ( 17 \sqrt { 17 } - 16 \sqrt { 10 } ) } { 27 }$ 
A)  $\frac { 5 ( 17 \sqrt { 85 } + 16 \sqrt { 10 } ) } { 27 }$ 
B)  $\frac { 5 ( 17 \sqrt { 85 } - 16 \sqrt { 10 } ) } { 27 }$ 
C)  $\frac { 5 ( 85 \sqrt { 17 } - 16 \sqrt { 10 } ) } { 27 }$ 
D)  $\frac { 5 ( 85 \sqrt { 17 } + 16 \sqrt { 10 } ) } { 27 }$ 
E)  $\frac { 5 ( 17 \sqrt { 17 } - 16 \sqrt { 10 } ) } { 27 }$

Question 14

A cylindrical sewer pipe of radius $\frac { 5 } { 2 }$ feet is half full of water of density 62.5 pounds per cubic foot. A gate used to seal off the sewer is placed perpendicular to the pipe opening. The force, in pounds, due to hydrostatic pressure on one end of the gate is

Accepted Answer

A)  $\frac { 1,125 } { 8 }$ 
B)  $\frac { 500 } { 3 }$ 
C)  $1,125$ 
D)  $\frac { 15,625 } { 24 }$ 
E)  $\frac { 125 } { 24 }$ 
A)  $\frac { 1,125 } { 8 }$ 
B)  $\frac { 500 } { 3 }$ 
C)  $1,125$ 
D)  $\frac { 15,625 } { 24 }$ 
E)  $\frac { 125 } { 24 }$

Question 15

Let $\bar { x }$ denote the x-coordinate of the centroid of the region enclosed by and $y ^ { 2 } = 4 x , y = 0,$ $x = 1 \text {, }$ and x=4 Then $\bar { x }$ =

Accepted Answer

A)  $\frac { 1,125 } { 8 }$ 
B)  $\frac { 1 } { 3 }$ 
C)  $\frac { 93 } { 35 }$ 
D)  $\frac { 3 } { 4 }$ 
E)  $\frac { 125 } { 24 }$ 
A)  $\frac { 1,125 } { 8 }$ 
B)  $\frac { 1 } { 3 }$ 
C)  $\frac { 93 } { 35 }$ 
D)  $\frac { 3 } { 4 }$ 
E)  $\frac { 125 } { 24 }$

Question 16

Let V be the volume of the solid generated by revolving the region enclosed by about the y-axis. Then V is

Accepted Answer

A)  $\frac { \pi } { 3 }$ 
B)  $\frac { 704 \pi } { 5 }$ 
C)  $\frac { 176 \pi } { 15 }$ 
D)  $\frac { 8 \pi } { 3 }$ 
E)  $\pi$ 
A)  $\frac { \pi } { 3 }$ 
B)  $\frac { 704 \pi } { 5 }$ 
C)  $\frac { 176 \pi } { 15 }$ 
D)  $\frac { 8 \pi } { 3 }$ 
E)  $\pi$

Question 17

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = -3 to x = 3. The cross-sections of this solid perpendicular to the x-axis run from $y = - \sqrt { 9 - x ^ { 2 } }$ to $y = \sqrt { 9 - x ^ { 2 } }$ and they are squares with bases in the xy-plane. Then V is

Accepted Answer

A)  $\frac { 25 } { 3 }$ 
B) 144 
C)  $\frac { 256 } { 3 }$ 
D)  $72$ 
E)  $55$ 
A)  $\frac { 25 } { 3 }$ 
B) 144 
C)  $\frac { 256 } { 3 }$ 
D)  $72$ 
E)  $55$

Question 18

Let $\bar { y }$ denote the y-coordinate of the centroid of the region enclosed by $x = \frac { y ^ { 2 } } { 4 }$ and $x = 4 .$ Then $\bar { y }$ =

Accepted Answer

A) 0 
B)  $\frac { 1 } { 5 }$ 
C) 1 
D)  $\frac { 3 } { 4 }$ 
E)  $\frac { 45 } { 28 }$ 
A) 0 
B)  $\frac { 1 } { 5 }$ 
C) 1 
D)  $\frac { 3 } { 4 }$ 
E)  $\frac { 45 } { 28 }$

Question 19

The arc length of $y = - \frac { 2 } { 3 } ( x - 3 ) ^ { \frac { 3 } { 2 } } + 2 \text { for } x \in [ 4,7 ]$ is

Accepted Answer

A)  $\frac { 2 ( 5 \sqrt { 5 } - 2 \sqrt { 2 } ) } { 3 }$ 
B)  $\frac { 3 ( 5 \sqrt { 5 } - 2 \sqrt { 2 } ) } { 2 }$ 
C)  $\frac { 2 ( 5 \sqrt { 5 } - 3 \sqrt { 2 } ) } { 3 }$ 
D)  $\frac { 3 ( 5 \sqrt { 5 } - 3 \sqrt { 2 } ) } { 2 }$ 
E)  $\frac { 2 ( 5 \sqrt { 5 } + 2 \sqrt { 2 } ) } { 3 }$ 
A)  $\frac { 2 ( 5 \sqrt { 5 } - 2 \sqrt { 2 } ) } { 3 }$ 
B)  $\frac { 3 ( 5 \sqrt { 5 } - 2 \sqrt { 2 } ) } { 2 }$ 
C)  $\frac { 2 ( 5 \sqrt { 5 } - 3 \sqrt { 2 } ) } { 3 }$ 
D)  $\frac { 3 ( 5 \sqrt { 5 } - 3 \sqrt { 2 } ) } { 2 }$ 
E)  $\frac { 2 ( 5 \sqrt { 5 } + 2 \sqrt { 2 } ) } { 3 }$

Question 20

A spring in equilibrium is 0.8 meters long and has a spring constant of 5 Newtons/meter. The work done by the spring when it is compressed from equilibrium to a length of 0.5 meters, in Newton-meters, is
​

Accepted Answer

A) -0.225 
B) -0.365 
C) -0.375 
D) -0.425 
E) -0.385 
A) -0.225 
B) -0.365 
C) -0.375 
D) -0.425 
E) -0.385

Let V be the volume of the solid generated by revolving the region enclosed by $y = \sqrt { x },$ the x-axis, x = 4; about $y = 3$ . Then V is

Let V be the volume of the solid generated by revolving the region enclosed by y = sec x, the x-axis, $x = \frac { \pi } { 3 };$ about the x-axis. Then V is

A cylindrical sewer pipe of radius $\frac { 1 } { 2 }$ foot is half full of water of density 62.5 pounds per cubic foot. A gate used to seal off the sewer is placed perpendicular to the pipe opening. The force, in pounds, due to hydrostatic pressure on one end of the gate is

Three particles having masses 1, 2, and 3 kg are located on the x-axis at -1,2,3 respectively. The center of mass of this system is located at x =

Which of these integrals can be used to calculate the surface area of the solid formed by revolving the region bounded by $f ( x ) = x ^ { 4 } - 3 x,$ the x-axis, x = -2, and x = 0 about the x-axis?

Let V be the volume of the solid generated by revolving the region enclosed by about the x-axis. Then V is

By the Pappus Theorem, the volume of the solid formed by revolving the region enclosed by the circle $( x - 3 ) ^ { 2 } + y ^ { 2 } = 1$ about the y-axis is

The arc length of $y = \frac { x ^ { \frac { 3 } { 2 } } } { 3 } - x ^ { \frac { 1 } { 2 } } \text { for } x \in [ 1,4 ]$ is

Let $\bar { x }$ denote the x-coordinate of the centroid of the region enclosed by $y = x ^ { 2 }$ and $y = 1$ Then $\bar { x }$ =

A cylindrical sewer pipe of radius $\frac { 1 } { 2 }$ feet is half full of water of density 62.5 pounds per cubic foot. A gate used to seal off the sewer is placed perpendicular to the pipe opening. The force, in pounds, due to hydrostatic pressure on one end of the gate is

Let A denote the area enclosed by the equations $y ^ { 2 } = x$ and $y = 2 - x$ Then A is

The arc length of $y = x ^ { \frac { 3 } { 2 } } \text { for } x \in [ 1,8 ]$ is

The arc length of $y = - ( x - 1 ) ^ { \frac { 3 } { 2 } } - 1 \text { for } x \in [ 5,10 ]$ is

A cylindrical sewer pipe of radius $\frac { 5 } { 2 }$ feet is half full of water of density 62.5 pounds per cubic foot. A gate used to seal off the sewer is placed perpendicular to the pipe opening. The force, in pounds, due to hydrostatic pressure on one end of the gate is

Let $\bar { x }$ denote the x-coordinate of the centroid of the region enclosed by and $y ^ { 2 } = 4 x , y = 0,$ $x = 1 \text {, }$ and x=4 Then $\bar { x }$ =

Let V be the volume of the solid generated by revolving the region enclosed by about the y-axis. Then V is

Let $\bar { y }$ denote the y-coordinate of the centroid of the region enclosed by $x = \frac { y ^ { 2 } } { 4 }$ and $x = 4 .$ Then $\bar { y }$ =

The arc length of $y = - \frac { 2 } { 3 } ( x - 3 ) ^ { \frac { 3 } { 2 } } + 2 \text { for } x \in [ 4,7 ]$ is

A spring in equilibrium is 0.8 meters long and has a spring constant of 5 Newtons/meter. The work done by the spring when it is compressed from equilibrium to a length of 0.5 meters, in Newton-meters, is

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Exam 7: Applications of the Integral

Let V be the volume of the solid generated by revolving the region enclosed by y=x,y = \sqrt { x },y=x​, the x-axis, x = 4; about y=3y = 3y=3 . Then V is

Let V be the volume of the solid generated by revolving the region enclosed by y = sec x, the x-axis, x=π3;x = \frac { \pi } { 3 };x=3π​; about the x-axis. Then V is

A cylindrical sewer pipe of radius 12\frac { 1 } { 2 }21​ foot is half full of water of density 62.5 pounds per cubic foot. A gate used to seal off the sewer is placed perpendicular to the pipe opening. The force, in pounds, due to hydrostatic pressure on one end of the gate is

Three particles having masses 1, 2, and 3 kg are located on the x-axis at -1,2,3 respectively. The center of mass of this system is located at x =

Which of these integrals can be used to calculate the surface area of the solid formed by revolving the region bounded by f(x)=x4−3x,f ( x ) = x ^ { 4 } - 3 x,f(x)=x4−3x, the x-axis, x = -2, and x = 0 about the x-axis?

Let V be the volume of the solid generated by revolving the region enclosed by about the x-axis. Then V is

By the Pappus Theorem, the volume of the solid formed by revolving the region enclosed by the circle (x−3)2+y2=1( x - 3 ) ^ { 2 } + y ^ { 2 } = 1(x−3)2+y2=1 about the y-axis is

The arc length of y=x323−x12 for x∈[1,4]y = \frac { x ^ { \frac { 3 } { 2 } } } { 3 } - x ^ { \frac { 1 } { 2 } } \text { for } x \in [ 1,4 ]y=3x23​​−x21​ for x∈[1,4] is

Let xˉ\bar { x }xˉ denote the x-coordinate of the centroid of the region enclosed by y=x2y = x ^ { 2 }y=x2 and y=1y = 1y=1 Then xˉ\bar { x }xˉ =

A cylindrical sewer pipe of radius 12\frac { 1 } { 2 }21​ feet is half full of water of density 62.5 pounds per cubic foot. A gate used to seal off the sewer is placed perpendicular to the pipe opening. The force, in pounds, due to hydrostatic pressure on one end of the gate is

Let A denote the area enclosed by the equations y2=xy ^ { 2 } = xy2=x and y=2−xy = 2 - xy=2−x Then A is

The arc length of y=x32 for x∈[1,8]y = x ^ { \frac { 3 } { 2 } } \text { for } x \in [ 1,8 ]y=x23​ for x∈[1,8] is

The arc length of y=−(x−1)32−1 for x∈[5,10]y = - ( x - 1 ) ^ { \frac { 3 } { 2 } } - 1 \text { for } x \in [ 5,10 ]y=−(x−1)23​−1 for x∈[5,10] is

A cylindrical sewer pipe of radius 52\frac { 5 } { 2 }25​ feet is half full of water of density 62.5 pounds per cubic foot. A gate used to seal off the sewer is placed perpendicular to the pipe opening. The force, in pounds, due to hydrostatic pressure on one end of the gate is

Let xˉ\bar { x }xˉ denote the x-coordinate of the centroid of the region enclosed by and y2=4x,y=0,y ^ { 2 } = 4 x , y = 0,y2=4x,y=0, x=1, x = 1 \text {, }x=1, and x=4 Then xˉ\bar { x }xˉ =

Let V be the volume of the solid generated by revolving the region enclosed by about the y-axis. Then V is

Let yˉ\bar { y }yˉ​ denote the y-coordinate of the centroid of the region enclosed by x=y24x = \frac { y ^ { 2 } } { 4 }x=4y2​ and x=4.x = 4 .x=4. Then yˉ\bar { y }yˉ​ =

The arc length of y=−23(x−3)32+2 for x∈[4,7]y = - \frac { 2 } { 3 } ( x - 3 ) ^ { \frac { 3 } { 2 } } + 2 \text { for } x \in [ 4,7 ]y=−32​(x−3)23​+2 for x∈[4,7] is

A spring in equilibrium is 0.8 meters long and has a spring constant of 5 Newtons/meter. The work done by the spring when it is compressed from equilibrium to a length of 0.5 meters, in Newton-meters, is ​

Preparing for Calculus

Limits and Continuity

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Techniques of Integration

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Let V be the volume of the solid generated by revolving the region enclosed by $y = \sqrt { x },$ the x-axis, x = 4; about $y = 3$ . Then V is

Let V be the volume of the solid generated by revolving the region enclosed by y = sec x, the x-axis, $x = \frac { \pi } { 3 };$ about the x-axis. Then V is

A cylindrical sewer pipe of radius $\frac { 1 } { 2 }$ foot is half full of water of density 62.5 pounds per cubic foot. A gate used to seal off the sewer is placed perpendicular to the pipe opening. The force, in pounds, due to hydrostatic pressure on one end of the gate is

Which of these integrals can be used to calculate the surface area of the solid formed by revolving the region bounded by $f ( x ) = x ^ { 4 } - 3 x,$ the x-axis, x = -2, and x = 0 about the x-axis?

By the Pappus Theorem, the volume of the solid formed by revolving the region enclosed by the circle $( x - 3 ) ^ { 2 } + y ^ { 2 } = 1$ about the y-axis is

The arc length of $y = \frac { x ^ { \frac { 3 } { 2 } } } { 3 } - x ^ { \frac { 1 } { 2 } } \text { for } x \in [ 1,4 ]$ is

Let $\bar { x }$ denote the x-coordinate of the centroid of the region enclosed by $y = x ^ { 2 }$ and $y = 1$ Then $\bar { x }$ =

A cylindrical sewer pipe of radius $\frac { 1 } { 2 }$ feet is half full of water of density 62.5 pounds per cubic foot. A gate used to seal off the sewer is placed perpendicular to the pipe opening. The force, in pounds, due to hydrostatic pressure on one end of the gate is

Let A denote the area enclosed by the equations $y ^ { 2 } = x$ and $y = 2 - x$ Then A is

The arc length of $y = x ^ { \frac { 3 } { 2 } } \text { for } x \in [ 1,8 ]$ is

The arc length of $y = - ( x - 1 ) ^ { \frac { 3 } { 2 } } - 1 \text { for } x \in [ 5,10 ]$ is

A cylindrical sewer pipe of radius $\frac { 5 } { 2 }$ feet is half full of water of density 62.5 pounds per cubic foot. A gate used to seal off the sewer is placed perpendicular to the pipe opening. The force, in pounds, due to hydrostatic pressure on one end of the gate is

Let $\bar { x }$ denote the x-coordinate of the centroid of the region enclosed by and $y ^ { 2 } = 4 x , y = 0,$ $x = 1 \text {, }$ and x=4 Then $\bar { x }$ =

Let $\bar { y }$ denote the y-coordinate of the centroid of the region enclosed by $x = \frac { y ^ { 2 } } { 4 }$ and $x = 4 .$ Then $\bar { y }$ =

The arc length of $y = - \frac { 2 } { 3 } ( x - 3 ) ^ { \frac { 3 } { 2 } } + 2 \text { for } x \in [ 4,7 ]$ is

A spring in equilibrium is 0.8 meters long and has a spring constant of 5 Newtons/meter. The work done by the spring when it is compressed from equilibrium to a length of 0.5 meters, in Newton-meters, is