Question 1

Let V be the volume of the solid generated by revolving the region enclosed by x = -y3, x = y3 - 2y2; about the x-axis. Then V is

Accepted Answer

A)  $\frac { \pi } { 3 }$ 
B)  $\frac { \pi } { 5 }$ 
C)  $24 \pi$ 
D)  $\frac { 1088 \pi } { 15 }$ 
E)  $\frac { 8 \pi } { 5 }$ 
A)  $\frac { \pi } { 3 }$ 
B)  $\frac { \pi } { 5 }$ 
C)  $24 \pi$ 
D)  $\frac { 1088 \pi } { 15 }$ 
E)  $\frac { 8 \pi } { 5 }$

Question 2

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

Accepted Answer

A) 1 
B)  $\frac { 2 } { 5 }$ 
C)  $\frac { 12 } { 5 }$ 
D)  $\frac { 3 } { 4 }$ 
E) 0 
A) 1 
B)  $\frac { 2 } { 5 }$ 
C)  $\frac { 12 } { 5 }$ 
D)  $\frac { 3 } { 4 }$ 
E) 0

Question 3

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

Accepted Answer

A)  $8 \pi$ 
B)  $7 \pi$ 
C)  $6 \pi$ 
D)  $5 \pi$ 
E)  $4 \pi$ 
A)  $8 \pi$ 
B)  $7 \pi$ 
C)  $6 \pi$ 
D)  $5 \pi$ 
E)  $4 \pi$

Question 4

Let V be the volume of the solid generated by revolving the region bounded above by $y = 2 x ^ { 2 }$ and bounded below the x-axis, $x \in [ 0 , \sqrt { 2 } ];$ about the y-axis. Then V is

Accepted Answer

A)  $4 \pi$ 
B)  $\frac { 128 \pi } { 3 }$ 
C)  $\frac { 32 \pi } { 5 }$ 
D)  $\frac { 8 \pi } { 3 }$ 
E)  $\frac { 8 \pi } { 5 }$ 
A)  $4 \pi$ 
B)  $\frac { 128 \pi } { 3 }$ 
C)  $\frac { 32 \pi } { 5 }$ 
D)  $\frac { 8 \pi } { 3 }$ 
E)  $\frac { 8 \pi } { 5 }$

Question 5

By the Pappus Theorem, the volume of the solid formed by revolving the region enclosed by the circle $x ^ { 2 } + ( y - 5 ) ^ { 2 } = 1$ about the x-axis is

Accepted Answer

A)  $4 \pi ^ { 2 }$ 
B)  $10 \pi ^ { 2 }$ 
C)  $36 \pi ^ { 2 }$ 
D)  $24 \pi ^ { 2 }$ 
E)  $6 \pi ^ { 2 }$ 
A)  $4 \pi ^ { 2 }$ 
B)  $10 \pi ^ { 2 }$ 
C)  $36 \pi ^ { 2 }$ 
D)  $24 \pi ^ { 2 }$ 
E)  $6 \pi ^ { 2 }$

Question 6

The arc length of $y = - \frac { 2 } { 3 } ( x + 3 ) ^ { \frac { 3 } { 2 } } \text { for } x \in [ - 2,1 ]$ 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 7

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 3. The cross-sections of this solid perpendicular to the x-axis run from $y = \frac { x ^ { 2 } } { 9 }$ to $y = \sqrt { \frac { x } { 3 } }$ and they are equilateral triangles with bases in the xy-plane. Then V is

Accepted Answer

A)  $\frac { 27 \sqrt { 3 } } { 280 }$ 
B)  $\frac { 27 \sqrt { 3 } } { 140 }$ 
C)  $\frac { 27 \sqrt { 3 } } { 70 }$ 
D)  $\frac { 27 \sqrt { 3 } } { 35 }$ 
E)  $\frac { 27 \sqrt { 3 } } { 350 }$ 
A)  $\frac { 27 \sqrt { 3 } } { 280 }$ 
B)  $\frac { 27 \sqrt { 3 } } { 140 }$ 
C)  $\frac { 27 \sqrt { 3 } } { 70 }$ 
D)  $\frac { 27 \sqrt { 3 } } { 35 }$ 
E)  $\frac { 27 \sqrt { 3 } } { 350 }$

Question 8

The work done by a winch winding in an 80-foot rope weighing 6 pounds per foot, in foot-pounds, is
​

Accepted Answer

A) 15,200 
B) 23,500 
C) 29,800 
D) 19,200 
E) 18,500 
A) 15,200 
B) 23,500 
C) 29,800 
D) 19,200 
E) 18,500

Question 9

Let V be the volume of the solid generated by revolving the region bounded above by $y = \sqrt { \cos x }$ and bounded below by the x-axis from x = 0 to $x = \frac { \pi } { 2 };$ about the x-axis. Then V is

Accepted Answer

A)  $\pi$ 
B)  $\frac { 3 \pi } { 2 }$ 
C)  $2 \pi$ 
D)  $\frac { 5 \pi } { 2 }$ 
E)  $3 \pi$ 
A)  $\pi$ 
B)  $\frac { 3 \pi } { 2 }$ 
C)  $2 \pi$ 
D)  $\frac { 5 \pi } { 2 }$ 
E)  $3 \pi$

Question 10

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

Accepted Answer

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

Question 11

A trough whose cross-section is a trapezoid whose lower base is 2 feet long and upper base is 4 feet long, and is 1 foot high is filled with water of density 62.5 pounds per cubic foot. The force, in pounds, due to hydrostatic pressure on one end of the trough is

Accepted Answer

A)  $3,000$ 
B)  $\frac { 500 } { 3 }$ 
C)  $\frac { 875 } { 6 }$ 
D)  $1,000$ 
E)  $\frac { 250 } { 3 }$ 
A)  $3,000$ 
B)  $\frac { 500 } { 3 }$ 
C)  $\frac { 875 } { 6 }$ 
D)  $1,000$ 
E)  $\frac { 250 } { 3 }$

Question 12

A spring in equilibrium is 12 centimeters long and an external force of 60 dynes compresses the spring to a length of 10 centimeters. The work done by the spring in compressing it from equilibrium to 8 centimeters, in dyne-centimeters, is
​

Accepted Answer

A) -264 
B) -240 
C) -238 
D) -212 
E) -200 
A) -264 
B) -240 
C) -238 
D) -212 
E) -200

Question 13

A trough whose cross-section is a right isosceles triangle whose base is 16 feet long and altitude is 4 feet long is filled with water of density 62.5 pounds per cubic foot. The force, in pounds, due to hydrostatic pressure on one end of the trough is

Accepted Answer

A)  $\frac { 8,000 } { 3 }$ 
B)  $\frac { 2,750 } { 3 }$ 
C)  $\frac { 2,550 } { 3 }$ 
D)  $\frac { 2,450 } { 3 }$ 
E)  $\frac { 2250 } { 3 }$ 
A)  $\frac { 8,000 } { 3 }$ 
B)  $\frac { 2,750 } { 3 }$ 
C)  $\frac { 2,550 } { 3 }$ 
D)  $\frac { 2,450 } { 3 }$ 
E)  $\frac { 2250 } { 3 }$

Question 14

Let A denote the area enclosed by the equations $x = y ^ { 2 }$ and $x = y + 6$ Then A is

Accepted Answer

A)  $\frac { 625 } { 6 }$ 
B)  $\frac { 225 } { 6 }$ 
C)  $\frac { 125 } { 6 }$ 
D)  $\frac { 25 } { 6 }$ 
E)  $\frac { 5 } { 6 }$ 
A)  $\frac { 625 } { 6 }$ 
B)  $\frac { 225 } { 6 }$ 
C)  $\frac { 125 } { 6 }$ 
D)  $\frac { 25 } { 6 }$ 
E)  $\frac { 5 } { 6 }$

Question 15

Let V be the volume of the solid generated by revolving the region enclosed by y = 5 - x2, y = 4; about the x-axis. Then V is

Accepted Answer

A)  $\frac { \pi } { 3 }$ 
B)  $\frac { 64 \pi } { 5 }$ 
C)  $\frac { 176 \pi } { 15 }$ 
D)  $\frac { \pi } { 5 }$ 
E)  $\frac { 8 \pi } { 5 }$ 
A)  $\frac { \pi } { 3 }$ 
B)  $\frac { 64 \pi } { 5 }$ 
C)  $\frac { 176 \pi } { 15 }$ 
D)  $\frac { \pi } { 5 }$ 
E)  $\frac { 8 \pi } { 5 }$

Question 16

A trough whose cross-section is a right isosceles triangle whose base is 6 feet long and altitude is 6 feet long is filled with water of density 62.5 pounds per cubic foot. The force, in pounds, due to hydrostatic pressure on one end of the trough is

Accepted Answer

A)  $500$ 
B)  $\frac { 2,000 } { 3 }$ 
C)  $1,250$ 
D)  $1,850$ 
E)  $2,250$ 
A)  $500$ 
B)  $\frac { 2,000 } { 3 }$ 
C)  $1,250$ 
D)  $1,850$ 
E)  $2,250$

Question 17

Let V be the volume of the solid generated by revolving the region enclosed by $y = 2 x ^ { 2 },$ the x-axis, $x = 1 ;$ about the x-axis. Then V is

Accepted Answer

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

Question 18

Let V be the volume of the solid generated by revolving the region enclosed by $y = \sqrt { x },$ the y-axis, $y = 2;$ about the y-axis. Then V is

Accepted Answer

A)  $\frac { 52 \pi } { 5 }$ 
B)  $\frac { 42 \pi } { 5 }$ 
C)  $\frac { 32 \pi } { 5 }$ 
D)  $\frac { 22 \pi } { 5 }$ 
E)  $\frac { 12 \pi } { 5 }$ 
A)  $\frac { 52 \pi } { 5 }$ 
B)  $\frac { 42 \pi } { 5 }$ 
C)  $\frac { 32 \pi } { 5 }$ 
D)  $\frac { 22 \pi } { 5 }$ 
E)  $\frac { 12 \pi } { 5 }$

Question 19

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 3. The cross-sections of this solid perpendicular to the x-axis run from $y = 0$ to $y = 3 - x$ and they are semicircles with diameters in the xy-plane. Then V is

Accepted Answer

A)  $\frac { 25 \pi } { 3 }$ 
B)  $\frac { 7 \pi } { 3 }$ 
C)  $\frac { 9 \pi } { 35 }$ 
D)  $\frac { 9 \pi } { 8 }$ 
E)  $\frac { 4 \pi } { 5 }$ 
A)  $\frac { 25 \pi } { 3 }$ 
B)  $\frac { 7 \pi } { 3 }$ 
C)  $\frac { 9 \pi } { 35 }$ 
D)  $\frac { 9 \pi } { 8 }$ 
E)  $\frac { 4 \pi } { 5 }$

Question 20

A trough whose cross-section is a trapezoid whose lower base is 2 feet long and upper base is 6 feet long, and is 1 foot high is filled with water of density 62.5 pounds per cubic foot. The force, in pounds, due to hydrostatic pressure on one end of the trough is

Accepted Answer

A)  $\frac { 625 } { 6 }$ 
B)  $\frac { 500 } { 3 }$ 
C)  $\frac { 875 } { 6 }$ 
D)  $1,000$ 
E)  $125$ 
A)  $\frac { 625 } { 6 }$ 
B)  $\frac { 500 } { 3 }$ 
C)  $\frac { 875 } { 6 }$ 
D)  $1,000$ 
E)  $125$

Let V be the volume of the solid generated by revolving the region enclosed by x = -y³, x = y³ - 2y²; about the x-axis. Then V is

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

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

Let V be the volume of the solid generated by revolving the region bounded above by $y = 2 x ^ { 2 }$ and bounded below the x-axis, $x \in [ 0 , \sqrt { 2 } ];$ about the y-axis. Then V is

By the Pappus Theorem, the volume of the solid formed by revolving the region enclosed by the circle $x ^ { 2 } + ( y - 5 ) ^ { 2 } = 1$ about the x-axis is

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

The work done by a winch winding in an 80-foot rope weighing 6 pounds per foot, in foot-pounds, is

Let V be the volume of the solid generated by revolving the region bounded above by $y = \sqrt { \cos x }$ and bounded below by the x-axis from x = 0 to $x = \frac { \pi } { 2 };$ about the x-axis. Then V is

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

A trough whose cross-section is a trapezoid whose lower base is 2 feet long and upper base is 4 feet long, and is 1 foot high is filled with water of density 62.5 pounds per cubic foot. The force, in pounds, due to hydrostatic pressure on one end of the trough is

A spring in equilibrium is 12 centimeters long and an external force of 60 dynes compresses the spring to a length of 10 centimeters. The work done by the spring in compressing it from equilibrium to 8 centimeters, in dyne-centimeters, is

A trough whose cross-section is a right isosceles triangle whose base is 16 feet long and altitude is 4 feet long is filled with water of density 62.5 pounds per cubic foot. The force, in pounds, due to hydrostatic pressure on one end of the trough is

Let A denote the area enclosed by the equations $x = y ^ { 2 }$ and $x = y + 6$ Then A is

Let V be the volume of the solid generated by revolving the region enclosed by y = 5 - x², y = 4; about the x-axis. Then V is

A trough whose cross-section is a right isosceles triangle whose base is 6 feet long and altitude is 6 feet long is filled with water of density 62.5 pounds per cubic foot. The force, in pounds, due to hydrostatic pressure on one end of the trough is

Let V be the volume of the solid generated by revolving the region enclosed by $y = 2 x ^ { 2 },$ the x-axis, $x = 1 ;$ about the x-axis. Then V is

Let V be the volume of the solid generated by revolving the region enclosed by $y = \sqrt { x },$ the y-axis, $y = 2;$ about the y-axis. Then V is

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 3. The cross-sections of this solid perpendicular to the x-axis run from $y = 0$ to $y = 3 - x$ and they are semicircles with diameters in the xy-plane. Then V is

A trough whose cross-section is a trapezoid whose lower base is 2 feet long and upper base is 6 feet long, and is 1 foot high is filled with water of density 62.5 pounds per cubic foot. The force, in pounds, due to hydrostatic pressure on one end of the trough 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 x = -y3, x = y3 - 2y2; about the x-axis. Then V is

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

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;x = 4;x=4; about the x-axis. Then V is

Let V be the volume of the solid generated by revolving the region bounded above by y=2x2y = 2 x ^ { 2 }y=2x2 and bounded below the x-axis, x∈[0,2];x \in [ 0 , \sqrt { 2 } ];x∈[0,2​]; about the y-axis. Then V is

By the Pappus Theorem, the volume of the solid formed by revolving the region enclosed by the circle x2+(y−5)2=1x ^ { 2 } + ( y - 5 ) ^ { 2 } = 1x2+(y−5)2=1 about the x-axis is

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

The work done by a winch winding in an 80-foot rope weighing 6 pounds per foot, in foot-pounds, is ​

Let V be the volume of the solid generated by revolving the region bounded above by y=cos⁡xy = \sqrt { \cos x }y=cosx​ and bounded below by the x-axis from x = 0 to x=π2;x = \frac { \pi } { 2 };x=2π​; about the x-axis. Then V is

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

A trough whose cross-section is a trapezoid whose lower base is 2 feet long and upper base is 4 feet long, and is 1 foot high is filled with water of density 62.5 pounds per cubic foot. The force, in pounds, due to hydrostatic pressure on one end of the trough is

A spring in equilibrium is 12 centimeters long and an external force of 60 dynes compresses the spring to a length of 10 centimeters. The work done by the spring in compressing it from equilibrium to 8 centimeters, in dyne-centimeters, is ​

A trough whose cross-section is a right isosceles triangle whose base is 16 feet long and altitude is 4 feet long is filled with water of density 62.5 pounds per cubic foot. The force, in pounds, due to hydrostatic pressure on one end of the trough is

Let A denote the area enclosed by the equations x=y2x = y ^ { 2 }x=y2 and x=y+6x = y + 6x=y+6 Then A is

Let V be the volume of the solid generated by revolving the region enclosed by y = 5 - x2, y = 4; about the x-axis. Then V is

A trough whose cross-section is a right isosceles triangle whose base is 6 feet long and altitude is 6 feet long is filled with water of density 62.5 pounds per cubic foot. The force, in pounds, due to hydrostatic pressure on one end of the trough is

Let V be the volume of the solid generated by revolving the region enclosed by y=2x2,y = 2 x ^ { 2 },y=2x2, the x-axis, x=1;x = 1 ;x=1; about the x-axis. Then V is

Let V be the volume of the solid generated by revolving the region enclosed by y=x,y = \sqrt { x },y=x​, the y-axis, y=2;y = 2;y=2; about the y-axis. Then V is

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 3. The cross-sections of this solid perpendicular to the x-axis run from y=0y = 0y=0 to y=3−xy = 3 - xy=3−x and they are semicircles with diameters in the xy-plane. Then V is

A trough whose cross-section is a trapezoid whose lower base is 2 feet long and upper base is 6 feet long, and is 1 foot high is filled with water of density 62.5 pounds per cubic foot. The force, in pounds, due to hydrostatic pressure on one end of the trough is

Preparing for Calculus

Limits and Continuity

The Derivative

More About Derivatives

Applications of the Derivative

The Integral

Techniques of Integration

Infinite Series

Parametric Equations; Polar Equations

Vectors; Lines, Planes, and Quadric Surfaces in Space

Vector Functions

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

Multiple Integrals

Vector Calculus

Differential Equations

Filters

Let V be the volume of the solid generated by revolving the region enclosed by x = -y³, x = y³ - 2y²; about the x-axis. Then V is

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

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

Let V be the volume of the solid generated by revolving the region bounded above by $y = 2 x ^ { 2 }$ and bounded below the x-axis, $x \in [ 0 , \sqrt { 2 } ];$ about the y-axis. Then V is

By the Pappus Theorem, the volume of the solid formed by revolving the region enclosed by the circle $x ^ { 2 } + ( y - 5 ) ^ { 2 } = 1$ about the x-axis is

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

The work done by a winch winding in an 80-foot rope weighing 6 pounds per foot, in foot-pounds, is

Let V be the volume of the solid generated by revolving the region bounded above by $y = \sqrt { \cos x }$ and bounded below by the x-axis from x = 0 to $x = \frac { \pi } { 2 };$ about the x-axis. Then V is

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

A spring in equilibrium is 12 centimeters long and an external force of 60 dynes compresses the spring to a length of 10 centimeters. The work done by the spring in compressing it from equilibrium to 8 centimeters, in dyne-centimeters, is

Let A denote the area enclosed by the equations $x = y ^ { 2 }$ and $x = y + 6$ Then A is

Let V be the volume of the solid generated by revolving the region enclosed by y = 5 - x², y = 4; about the x-axis. Then V is

Let V be the volume of the solid generated by revolving the region enclosed by $y = 2 x ^ { 2 },$ the x-axis, $x = 1 ;$ about the x-axis. Then V is

Let V be the volume of the solid generated by revolving the region enclosed by $y = \sqrt { x },$ the y-axis, $y = 2;$ about the y-axis. Then V is

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 3. The cross-sections of this solid perpendicular to the x-axis run from $y = 0$ to $y = 3 - x$ and they are semicircles with diameters in the xy-plane. Then V is