Question 1

Let A denote the area enclosed by the equation $y = x ^ { 2 } - 4 x + 3$ and the x-axis. Then A is

Accepted Answer

A)  $\frac { 4 } { 9 }$ 
B)  $\frac { 4 } { 5 }$ 
C)  $\frac { 4 } { 3 }$ 
D)  $\frac { 8 } { 3 }$ 
E)  $\frac { 2 } { 3 }$ 
A)  $\frac { 4 } { 9 }$ 
B)  $\frac { 4 } { 5 }$ 
C)  $\frac { 4 } { 3 }$ 
D)  $\frac { 8 } { 3 }$ 
E)  $\frac { 2 } { 3 }$

Question 2

Let A denote the area enclosed by the equations $y = x ^ { 2 }$ and $y = 8 - x ^ { 2 }$ Then A is

Accepted Answer

A)  $\frac { 4 } { 3 }$ 
B)  $\frac { 8 } { 3 }$ 
C)  $\frac { 16 } { 3 }$ 
D)  $\frac { 32 } { 3 }$ 
E)  $\frac { 64 } { 3 }$ 
A)  $\frac { 4 } { 3 }$ 
B)  $\frac { 8 } { 3 }$ 
C)  $\frac { 16 } { 3 }$ 
D)  $\frac { 32 } { 3 }$ 
E)  $\frac { 64 } { 3 }$

Question 3

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

Accepted Answer

A)  $\frac { 2 } { 3 } ( 5 \sqrt { 5 } - 2 \sqrt { 2 } )$ 
B)  $\frac { 2 } { 3 } ( 7 \sqrt { 5 } - 3 \sqrt { 2 } )$ 
C)  $\frac { 2 } { 3 } ( 5 \sqrt { 7 } - 2 \sqrt { 3 } )$ 
D)  $\frac { 2 } { 3 } ( 7 \sqrt { 3 } - 2 \sqrt { 2 } )$ 
E)  $\frac { 2 } { 3 } ( 5 \sqrt { 3 } - 2 \sqrt { 2 } )$ 
A)  $\frac { 2 } { 3 } ( 5 \sqrt { 5 } - 2 \sqrt { 2 } )$ 
B)  $\frac { 2 } { 3 } ( 7 \sqrt { 5 } - 3 \sqrt { 2 } )$ 
C)  $\frac { 2 } { 3 } ( 5 \sqrt { 7 } - 2 \sqrt { 3 } )$ 
D)  $\frac { 2 } { 3 } ( 7 \sqrt { 3 } - 2 \sqrt { 2 } )$ 
E)  $\frac { 2 } { 3 } ( 5 \sqrt { 3 } - 2 \sqrt { 2 } )$

Question 4

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 4. The cross-sections of this solid perpendicular to the x-axis run from $y = \frac { x ^ { 2 } } { 8 }$ to $y = \sqrt { x }$ and they are equilateral triangles with bases in the xy-plane. Then V is

Accepted Answer

A)  $\frac { 72 \sqrt { 3 } } { 35 }$ 
B)  $\frac { 36 \sqrt { 3 } } { 33 }$ 
C)  $\frac { 18 \sqrt { 3 } } { 33 }$ 
D)  $\frac { 18 \sqrt { 3 } } { 35 }$ 
E)  $\frac { 12 \sqrt { 3 } } { 35 }$ 
A)  $\frac { 72 \sqrt { 3 } } { 35 }$ 
B)  $\frac { 36 \sqrt { 3 } } { 33 }$ 
C)  $\frac { 18 \sqrt { 3 } } { 33 }$ 
D)  $\frac { 18 \sqrt { 3 } } { 35 }$ 
E)  $\frac { 12 \sqrt { 3 } } { 35 }$

Question 5

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)  $\frac { 5 ( 17 \sqrt { 85 } - 18 \sqrt { 10 } ) } { 27 }$ 
C)  $\frac { 5 ( 17 \sqrt { 85 } + 16 \sqrt { 10 } ) } { 27 }$. 
D)  $\frac { 5 ( 17 \sqrt { 85 } + 16 \sqrt { 10 } ) } { 27 }$ 
E)  $\frac { 5 ( 17 \sqrt { 85 } + 18 \sqrt { 10 } ) } { 27 }$ 
A)  $\frac { 5 ( 17 \sqrt { 85 } - 16 \sqrt { 10 } ) } { 27 }$ 
B)  $\frac { 5 ( 17 \sqrt { 85 } - 18 \sqrt { 10 } ) } { 27 }$ 
C)  $\frac { 5 ( 17 \sqrt { 85 } + 16 \sqrt { 10 } ) } { 27 }$. 
D)  $\frac { 5 ( 17 \sqrt { 85 } + 16 \sqrt { 10 } ) } { 27 }$ 
E)  $\frac { 5 ( 17 \sqrt { 85 } + 18 \sqrt { 10 } ) } { 27 }$

Question 6

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 x-axis. Then V is

Accepted Answer

A)  $\sqrt { 3 } \pi$ 
B)  $8 \pi$ 
C)  $\frac { 88 \pi } { 3 }$ 
D)  $\frac { 8 \pi } { 3 }$ 
E)  $\frac { 8 \pi } { 5 }$ 
A)  $\sqrt { 3 } \pi$ 
B)  $8 \pi$ 
C)  $\frac { 88 \pi } { 3 }$ 
D)  $\frac { 8 \pi } { 3 }$ 
E)  $\frac { 8 \pi } { 5 }$

Question 7

The arc length of $8 y = x ^ { 4 } + \frac { 2 } { x ^ { 2 } } \text { for } x \in [ 1,2 ]$ is

Accepted Answer

A)  $\frac { 63 } { 16 }$ 
B)  $\frac { 53 } { 16 }$ 
C)  $\frac { 33 } { 16 }$ 
D)  $\frac { 23 } { 16 }$ 
E)  $\frac { 13 } { 16 }$ 
A)  $\frac { 63 } { 16 }$ 
B)  $\frac { 53 } { 16 }$ 
C)  $\frac { 33 } { 16 }$ 
D)  $\frac { 23 } { 16 }$ 
E)  $\frac { 13 } { 16 }$

Question 8

Let A denote the area enclosed by the equations $y = \sqrt [ 3 ] { x }$ and $y = 4 x$ Then A is

Accepted Answer

A)  $\frac { 1 } { 64 }$ 
B)  $\frac { 1 } { 32 }$ 
C)  $\frac { 1 } { 16 }$ 
D)  $\frac { 1 } { 8 }$ 
E)  $\frac { 1 } { 4 }$ 
A)  $\frac { 1 } { 64 }$ 
B)  $\frac { 1 } { 32 }$ 
C)  $\frac { 1 } { 16 }$ 
D)  $\frac { 1 } { 8 }$ 
E)  $\frac { 1 } { 4 }$

Question 9

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 2. The cross-sections of this solid perpendicular to the x-axis run from y = 0 to y = x2 and they are semicircles with bases in the xy-plane. Then V is

Accepted Answer

A)  $\frac { 2187 \pi } { 560 }$ 
B)  $\frac { 25 \pi } { 3 }$ 
C)  $\frac { 7 \pi } { 3 }$ 
D)  $\frac { 9 \pi } { 35 }$ 
E)  $\frac { 4 \pi } { 5 }$ 
A)  $\frac { 2187 \pi } { 560 }$ 
B)  $\frac { 25 \pi } { 3 }$ 
C)  $\frac { 7 \pi } { 3 }$ 
D)  $\frac { 9 \pi } { 35 }$ 
E)  $\frac { 4 \pi } { 5 }$

Question 10

A spring in equilibrium is 4 meters long and an external force of 8 Newtons stretches the spring to a length of 4.5 meters. The work done by the spring in stretching it from 4.5 meters to 5 meters, in Newton-meters, is
​

Accepted Answer

A) -22 
B) -17 
C) -13 
D) -9 
E) -6 
A) -22 
B) -17 
C) -13 
D) -9 
E) -6

Question 11

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

Accepted Answer

A)  $\frac { \pi } { 3 }$ 
B)  $\frac { 704 \pi } { 5 }$ 
C)  $\frac { 176 \pi } { 15 }$ 
D)  $\frac { 7 \pi } { 3 }$ 
E)  $\pi$ 
A)  $\frac { \pi } { 3 }$ 
B)  $\frac { 704 \pi } { 5 }$ 
C)  $\frac { 176 \pi } { 15 }$ 
D)  $\frac { 7 \pi } { 3 }$ 
E)  $\pi$

Question 12

Let A denote the area enclosed by the equations $y = 2 x ^ { 3 } - 3 x ^ { 2 } - 5 x$ and $y = x ^ { 3 } - 2 x ^ { 2 } - 3 x$ Then A is

Accepted Answer

A) 2 
B)  $\frac { 27 } { 12 }$ 
C) 3 
D)  $\frac { 37 } { 12 }$ 
E)  $\frac { 39 } { 12 }$ 
A) 2 
B)  $\frac { 27 } { 12 }$ 
C) 3 
D)  $\frac { 37 } { 12 }$ 
E)  $\frac { 39 } { 12 }$

Question 13

The arc length of $y = - \frac { 2 } { 3 } ( x + 3 ) ^ { \frac { 3 } { 2 } } + 1 \text { for } x \in [ - 2,1 ]$ 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 14

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 squares with bases in the xy-plane. Then V is

Accepted Answer

A)  $\frac { 72 } { 35 }$ 
B) 14 
C)  $\frac { 56 } { 3 }$ 
D) 9 
E)  $\frac { 32 } { 5 }$ 
A)  $\frac { 72 } { 35 }$ 
B) 14 
C)  $\frac { 56 } { 3 }$ 
D) 9 
E)  $\frac { 32 } { 5 }$

Question 15

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 4. The cross-sections of this solid perpendicular to the x-axis run from $y = \frac { x ^ { 2 } } { 8 }$ to $y = \sqrt { x }$ and they are semicircles with diameters in the xy-plane. Then V is

Accepted Answer

A)  $\frac { \pi } { 3 }$ 
B)  $\frac { 47 \pi } { 3 }$ 
C)  $\frac { 25 \pi } { 3 }$ 
D)  $\frac { 7 \pi } { 3 }$ 
E)  $\frac { 9 \pi } { 35 }$ 
A)  $\frac { \pi } { 3 }$ 
B)  $\frac { 47 \pi } { 3 }$ 
C)  $\frac { 25 \pi } { 3 }$ 
D)  $\frac { 7 \pi } { 3 }$ 
E)  $\frac { 9 \pi } { 35 }$

Question 16

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

Accepted Answer

A)  $\frac { 64 \sqrt { 3 } } { 5 }$ 
B)  $\frac { 56 \sqrt { 3 } } { 5 }$ 
C)  $\frac { 48 \sqrt { 3 } } { 5 }$ 
D)  $\frac { 32 \sqrt { 3 } } { 5 }$ 
E)  $\frac { 8 \sqrt { 3 } } { 5 }$ 
A)  $\frac { 64 \sqrt { 3 } } { 5 }$ 
B)  $\frac { 56 \sqrt { 3 } } { 5 }$ 
C)  $\frac { 48 \sqrt { 3 } } { 5 }$ 
D)  $\frac { 32 \sqrt { 3 } } { 5 }$ 
E)  $\frac { 8 \sqrt { 3 } } { 5 }$

Question 17

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

Accepted Answer

A)  $\frac { \pi } { 3 }$ 
B)  $\frac { 72 \pi } { 5 }$ 
C)  $24 \pi$ 
D)  $\frac { 1088 \pi } { 15 }$ 
E)  $\frac { 8 \pi } { 5 }$ 
A)  $\frac { \pi } { 3 }$ 
B)  $\frac { 72 \pi } { 5 }$ 
C)  $24 \pi$ 
D)  $\frac { 1088 \pi } { 15 }$ 
E)  $\frac { 8 \pi } { 5 }$

Question 18

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

Accepted Answer

A)  $\frac { 10 \pi } { 3 }$ 
B)  $\frac { 11 \pi } { 5 }$ 
C)  $\frac { 9 \pi } { 5 }$ 
D)  $\frac { 8 \pi } { 3 }$ 
E)  $\frac { 8 \pi } { 5 }$ 
A)  $\frac { 10 \pi } { 3 }$ 
B)  $\frac { 11 \pi } { 5 }$ 
C)  $\frac { 9 \pi } { 5 }$ 
D)  $\frac { 8 \pi } { 3 }$ 
E)  $\frac { 8 \pi } { 5 }$

Question 19

A trough cross-section is a 2-foot-high and 4-foot-wide rectangle. If the trough 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)  $2,500$ 
C)  $3,200$ 
D)  $3,960$ 
E)  $4,500$ 
A)  $500$ 
B)  $2,500$ 
C)  $3,200$ 
D)  $3,960$ 
E)  $4,500$

Question 20

Let A denote the area enclosed by the equations $x = 3 y - y ^ { 2 }$ and x = y. Then A is

Accepted Answer

A)  $\frac { 40 } { 3 }$ 
B)  $\frac { 35 } { 3 }$ 
C)  $\frac { 25 } { 3 }$ 
D) 10 
E)  $\frac { 4 } { 3 }$ 
A)  $\frac { 40 } { 3 }$ 
B)  $\frac { 35 } { 3 }$ 
C)  $\frac { 25 } { 3 }$ 
D) 10 
E)  $\frac { 4 } { 3 }$

Let A denote the area enclosed by the equation $y = x ^ { 2 } - 4 x + 3$ and the x-axis. Then A is

Let A denote the area enclosed by the equations $y = x ^ { 2 }$ and $y = 8 - x ^ { 2 }$ Then A is

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

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

The arc length of $8 y = x ^ { 4 } + \frac { 2 } { x ^ { 2 } } \text { for } x \in [ 1,2 ]$ is

Let A denote the area enclosed by the equations $y = \sqrt [ 3 ] { x }$ and $y = 4 x$ Then A is

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 2. The cross-sections of this solid perpendicular to the x-axis run from y = 0 to y = x² and they are semicircles with bases in the xy-plane. Then V is

A spring in equilibrium is 4 meters long and an external force of 8 Newtons stretches the spring to a length of 4.5 meters. The work done by the spring in stretching it from 4.5 meters to 5 meters, in Newton-meters, is

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

Let A denote the area enclosed by the equations $y = 2 x ^ { 3 } - 3 x ^ { 2 } - 5 x$ and $y = x ^ { 3 } - 2 x ^ { 2 } - 3 x$ Then A is

The arc length of $y = - \frac { 2 } { 3 } ( x + 3 ) ^ { \frac { 3 } { 2 } } + 1 \text { for } x \in [ - 2,1 ]$ 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 squares with bases in the xy-plane. 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 = 4. The cross-sections of this solid perpendicular to the x-axis run from $y = \frac { x ^ { 2 } } { 8 }$ to $y = \sqrt { x }$ and they are semicircles with diameters in the xy-plane. 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 = 2. The cross-sections of this solid perpendicular to the x-axis run from $y = 0$ to $y = x ^ { 2 }$ and they are equilateral triangles with bases in the xy-plane. Then V is

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

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

A trough cross-section is a 2-foot-high and 4-foot-wide rectangle. If the trough 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 $x = 3 y - y ^ { 2 }$ and x = y. Then A is

Preparing for Calculus

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Exam 7: Applications of the Integral

Let A denote the area enclosed by the equation y=x2−4x+3y = x ^ { 2 } - 4 x + 3y=x2−4x+3 and the x-axis. Then A is

Let A denote the area enclosed by the equations y=x2y = x ^ { 2 }y=x2 and y=8−x2y = 8 - x ^ { 2 }y=8−x2 Then A is

The arc length of y=23x32 for x∈[1,4]y = \frac { 2 } { 3 } x ^ { \frac { 3 } { 2 } } \text { for } x \in [ 1,4 ]y=32​x23​ for x∈[1,4] 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

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 x-axis. Then V is

The arc length of 8y=x4+2x2 for x∈[1,2]8 y = x ^ { 4 } + \frac { 2 } { x ^ { 2 } } \text { for } x \in [ 1,2 ]8y=x4+x22​ for x∈[1,2] is

Let A denote the area enclosed by the equations y=x3y = \sqrt [ 3 ] { x }y=3x​ and y=4xy = 4 xy=4x Then A is

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 2. The cross-sections of this solid perpendicular to the x-axis run from y = 0 to y = x2 and they are semicircles with bases in the xy-plane. Then V is

A spring in equilibrium is 4 meters long and an external force of 8 Newtons stretches the spring to a length of 4.5 meters. The work done by the spring in stretching it from 4.5 meters to 5 meters, in Newton-meters, is ​

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

Let A denote the area enclosed by the equations y=2x3−3x2−5xy = 2 x ^ { 3 } - 3 x ^ { 2 } - 5 xy=2x3−3x2−5x and y=x3−2x2−3xy = x ^ { 3 } - 2 x ^ { 2 } - 3 xy=x3−2x2−3x Then A is

The arc length of y=−23(x+3)32+1 for x∈[−2,1]y = - \frac { 2 } { 3 } ( x + 3 ) ^ { \frac { 3 } { 2 } } + 1 \text { for } x \in [ - 2,1 ]y=−32​(x+3)23​+1 for x∈[−2,1] 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 squares with bases in the xy-plane. 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 = 2. The cross-sections of this solid perpendicular to the x-axis run from y=0y = 0y=0 to y=x2y = x ^ { 2 }y=x2 and they are equilateral triangles with bases in the xy-plane. Then V is

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

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

A trough cross-section is a 2-foot-high and 4-foot-wide rectangle. If the trough 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=3y−y2x = 3 y - y ^ { 2 }x=3y−y2 and x = y. Then A 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 A denote the area enclosed by the equation $y = x ^ { 2 } - 4 x + 3$ and the x-axis. Then A is

Let A denote the area enclosed by the equations $y = x ^ { 2 }$ and $y = 8 - x ^ { 2 }$ Then A is

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

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

The arc length of $8 y = x ^ { 4 } + \frac { 2 } { x ^ { 2 } } \text { for } x \in [ 1,2 ]$ is

Let A denote the area enclosed by the equations $y = \sqrt [ 3 ] { x }$ and $y = 4 x$ Then A is

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 2. The cross-sections of this solid perpendicular to the x-axis run from y = 0 to y = x² and they are semicircles with bases in the xy-plane. Then V is

A spring in equilibrium is 4 meters long and an external force of 8 Newtons stretches the spring to a length of 4.5 meters. The work done by the spring in stretching it from 4.5 meters to 5 meters, in Newton-meters, is

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

Let A denote the area enclosed by the equations $y = 2 x ^ { 3 } - 3 x ^ { 2 } - 5 x$ and $y = x ^ { 3 } - 2 x ^ { 2 } - 3 x$ Then A is

The arc length of $y = - \frac { 2 } { 3 } ( x + 3 ) ^ { \frac { 3 } { 2 } } + 1 \text { for } x \in [ - 2,1 ]$ 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 squares with bases in the xy-plane. 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 = 2. The cross-sections of this solid perpendicular to the x-axis run from $y = 0$ to $y = x ^ { 2 }$ and they are equilateral triangles with bases in the xy-plane. Then V is

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

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

Let A denote the area enclosed by the equations $x = 3 y - y ^ { 2 }$ and x = y. Then A is