Question 1

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

Accepted Answer

A)  $\pi$ 
B)  $\frac { 128 \pi } { 3 }$ 
C)  $\frac { 128 \pi } { 5 }$ 
D)  $\frac { 128 \pi } { 3 }$. 
E)  $\frac { 64 \pi } { 5 }$ 
A)  $\pi$ 
B)  $\frac { 128 \pi } { 3 }$ 
C)  $\frac { 128 \pi } { 5 }$ 
D)  $\frac { 128 \pi } { 3 }$. 
E)  $\frac { 64 \pi } { 5 }$

Question 2

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)  $\frac { \pi } { 48 }$ 
B)  $8 \pi$ 
C)  $\frac { \pi } { 3 }$ 
D)  $\frac { 1088 \pi } { 15 }$ 
E)  $\frac { 72 \pi } { 5 }$ 
A)  $\frac { \pi } { 48 }$ 
B)  $8 \pi$ 
C)  $\frac { \pi } { 3 }$ 
D)  $\frac { 1088 \pi } { 15 }$ 
E)  $\frac { 72 \pi } { 5 }$

Question 3

Let $\bar { x }$ denote the x-coordinate of the centroid of the region enclosed by $y = x ^ { 2 }$ and $y = 2 x + 3$ Then $\bar { x }$ =

Accepted Answer

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

Question 4

By the Pappus Theorem, the volume of the solid formed by revolving the region enclosed by the circle $( x - 4 ) ^ { 2 } + y ^ { 2 } = 9$ about the y-axis is

Accepted Answer

A)  $4 \pi ^ { 2 }$ 
B)  $26 \pi ^ { 2 }$ 
C)  $36 \pi ^ { 2 }$ 
D)  $72 \pi ^ { 2 }$ 
E)  $6 \pi ^ { 2 }$ 
A)  $4 \pi ^ { 2 }$ 
B)  $26 \pi ^ { 2 }$ 
C)  $36 \pi ^ { 2 }$ 
D)  $72 \pi ^ { 2 }$ 
E)  $6 \pi ^ { 2 }$

Question 5

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)  $\frac { 47 \pi } { 3 }$ 
B)  $\frac { 25 \pi } { 3 }$ 
C)  $\frac { 7 \pi } { 3 }$ 
D)  $\frac { 9 \pi } { 35 }$ 
E)  $\frac { 4 \pi } { 5 }$ 
A)  $\frac { 47 \pi } { 3 }$ 
B)  $\frac { 25 \pi } { 3 }$ 
C)  $\frac { 7 \pi } { 3 }$ 
D)  $\frac { 9 \pi } { 35 }$ 
E)  $\frac { 4 \pi } { 5 }$

Question 6

A trough cross-section is a 6-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)  $2,250$ 
B)  $3,500$ 
C)  $4,500$ 
D)  $5,000$ 
E)  $5,550$ 
A)  $2,250$ 
B)  $3,500$ 
C)  $4,500$ 
D)  $5,000$ 
E)  $5,550$

Question 7

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)  $\sqrt { 3 } \pi$ 
B)  $8 \pi$ 
C)  $24 \pi$ 
D)  $\frac { 1408 \pi } { 15 }$ 
E)  $\frac { 8 \pi } { 15 }$ 
A)  $\sqrt { 3 } \pi$ 
B)  $8 \pi$ 
C)  $24 \pi$ 
D)  $\frac { 1408 \pi } { 15 }$ 
E)  $\frac { 8 \pi } { 15 }$

Question 8

Let A denote the area enclosed by the equations $y = x ^ { 2 }$ and $y = 2 x$ 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 9

The arc length of $y = \frac { x ^ { 3 } } { 6 } + \frac { 1 } { 2 x } \text { for } x \in [ 1,4 ]$ is

Accepted Answer

A)  $\frac { 10 } { 3 }$ 
B)  $\frac { 87 } { 8 }$ 
C)  $\frac { 256 } { 3 }$ 
D)  $\frac { 55 } { 3 }$ 
E)  $\frac { 16 } { 5 }$ 
A)  $\frac { 10 } { 3 }$ 
B)  $\frac { 87 } { 8 }$ 
C)  $\frac { 256 } { 3 }$ 
D)  $\frac { 55 } { 3 }$ 
E)  $\frac { 16 } { 5 }$

Question 10

Let A denote the area enclosed by the equations $x = y ^ { 3 } - 2 y$ and $x = 7 y .$ Then A is

Accepted Answer

A)  $\frac { 81 } { 2 }$ 
B)  $\frac { 63 } { 2 }$ 
C)  $\frac { 45 } { 2 }$ 
D)  $\frac { 27 } { 2 }$ 
E)  $\frac { 9 } { 2 }$ 
A)  $\frac { 81 } { 2 }$ 
B)  $\frac { 63 } { 2 }$ 
C)  $\frac { 45 } { 2 }$ 
D)  $\frac { 27 } { 2 }$ 
E)  $\frac { 9 } { 2 }$

Question 11

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 $y = x ^ { 2 }$ to $y = 3 \sqrt { 3 x }$ and they are semicircles with diameters 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 12

The work done by a winch winding in a 50-meter chain weighing 3 Newtons per meter, in Newton-meters, is
​

Accepted Answer

A) 4,500 
B) 4,200 
C) 3,750 
D) 3,650 
E) 3,500 
A) 4,500 
B) 4,200 
C) 3,750 
D) 3,650 
E) 3,500

Question 13

Let V be the volume of the solid generated by revolving the region enclosed by about x = 2. Then V is

Accepted Answer

A)  $\frac { 48 \pi } { 5 }$ 
B)  $8 \pi$ 
C)  $24 \pi$ 
D)  $\frac { 1088 \pi } { 15 }$ 
E)  $\frac { 8 \pi } { 5 }$ 
A)  $\frac { 48 \pi } { 5 }$ 
B)  $8 \pi$ 
C)  $24 \pi$ 
D)  $\frac { 1088 \pi } { 15 }$ 
E)  $\frac { 8 \pi } { 5 }$

Question 14

Let V be the volume of the solid generated by revolving the region enclosed by y = x3, y = 2x2 - x3; about the y-axis. Then V is

Accepted Answer

A)  $\frac { \pi } { 48 }$ 
B)  $\frac { 64 \pi } { 5 }$ 
C)  $\frac { \pi } { 5 }$ 
D)  $\frac { 1088 \pi } { 15 }$ 
E)  $\frac { 8 \pi } { 5 }$ 
A)  $\frac { \pi } { 48 }$ 
B)  $\frac { 64 \pi } { 5 }$ 
C)  $\frac { \pi } { 5 }$ 
D)  $\frac { 1088 \pi } { 15 }$ 
E)  $\frac { 8 \pi } { 5 }$

Question 15

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

Accepted Answer

A)  $\frac { 27 } { 70 }$ 
B)  $\frac { 25 } { 3 }$ 
C)  $\frac { 256 } { 3 }$ 
D)  $\frac { 72 } { 35 }$ 
E)  $\frac { 32 } { 5 }$ 
A)  $\frac { 27 } { 70 }$ 
B)  $\frac { 25 } { 3 }$ 
C)  $\frac { 256 } { 3 }$ 
D)  $\frac { 72 } { 35 }$ 
E)  $\frac { 32 } { 5 }$

Question 16

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 $y = x ^ { 2 }$ to $y = 3 \sqrt { 3 x }$ and they are squares with bases in the xy-plane. Then V is

Accepted Answer

A)  $\frac { 2187 } { 70 }$ 
B) 144 
C)  $\frac { 256 } { 3 }$ 
D)  $\frac { 72 } { 35 }$ 
E)  $\frac { 32 } { 5 }$ 
A)  $\frac { 2187 } { 70 }$ 
B) 144 
C)  $\frac { 256 } { 3 }$ 
D)  $\frac { 72 } { 35 }$ 
E)  $\frac { 32 } { 5 }$

Question 17

The work done by a variable force $F ( x ) = x ^ { 2 } + 4$ Newtons that moves an object along a straight line in the direction of F from x = 1 meters to x = 3 meters, in Newton-meters, is

Accepted Answer

A)  $\frac { 50 } { 3 }$ 
B)  $\frac { 50 } { 7 }$ 
C)  $\frac { 55 } { 3 }$ 
D)  $\frac { 55 } { 7 }$ 
E)  $\frac { 64 } { 3 }$ 
A)  $\frac { 50 } { 3 }$ 
B)  $\frac { 50 } { 7 }$ 
C)  $\frac { 55 } { 3 }$ 
D)  $\frac { 55 } { 7 }$ 
E)  $\frac { 64 } { 3 }$

Question 18

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = -1 to x = 1. The cross-sections of this solid perpendicular to the x-axis run from $y = - 4 \sqrt { 1 - x ^ { 2 } }$ to $y = 4 \sqrt { 1 - x ^ { 2 } }$ and they are equilateral triangles with bases in the xy-plane. Then V is

Accepted Answer

A)  $\frac { 256 } { \sqrt { 3 } }$ 
B)  $\frac { 128 } { \sqrt { 3 } }$ 
C)  $\frac { 64 } { \sqrt { 3 } }$ 
D)  $\frac { 32 } { \sqrt { 3 } }$ 
E)  $\frac { 16 } { \sqrt { 3 } }$ 
A)  $\frac { 256 } { \sqrt { 3 } }$ 
B)  $\frac { 128 } { \sqrt { 3 } }$ 
C)  $\frac { 64 } { \sqrt { 3 } }$ 
D)  $\frac { 32 } { \sqrt { 3 } }$ 
E)  $\frac { 16 } { \sqrt { 3 } }$

Question 19

Let V be the volume of a wedge cut from a right-circular cylinder with a radius of 1 inch by two planes, one perpendicular to the axis of the cylinder and the other intersecting the first along a diameter of the circular plane section at an angle of $\frac { \pi } { 3 }$ radian. Then V is

Accepted Answer

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

Question 20

A trough whose cross-section is a trapezoid whose lower base is 4 feet long and upper base is 8 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)  $\frac { 10,125 } { 8 }$ 
D)  $1,000$ 
E)  $2,250$ 
A)  $3,000$ 
B)  $\frac { 500 } { 3 }$ 
C)  $\frac { 10,125 } { 8 }$ 
D)  $1,000$ 
E)  $2,250$

Let V be the volume of the solid generated by revolving the region enclosed by $y = \sqrt { x },$ the y-axis, y = 4; about y = 4. 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 $\bar { x }$ denote the x-coordinate of the centroid of the region enclosed by $y = x ^ { 2 }$ and $y = 2 x + 3$ Then $\bar { x }$ =

By the Pappus Theorem, the volume of the solid formed by revolving the region enclosed by the circle $( x - 4 ) ^ { 2 } + y ^ { 2 } = 9$ about the y-axis 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 trough cross-section is a 6-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 V be the volume of the solid generated by revolving the region enclosed by about the x-axis. Then V is

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

The arc length of $y = \frac { x ^ { 3 } } { 6 } + \frac { 1 } { 2 x } \text { for } x \in [ 1,4 ]$ is

Let A denote the area enclosed by the equations $x = y ^ { 3 } - 2 y$ and $x = 7 y .$ 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 = 3. The cross-sections of this solid perpendicular to the x-axis run from $y = x ^ { 2 }$ to $y = 3 \sqrt { 3 x }$ and they are semicircles with diameters in the xy-plane. Then V is

The work done by a winch winding in a 50-meter chain weighing 3 Newtons per meter, in Newton-meters, is

Let V be the volume of the solid generated by revolving the region enclosed by about x = 2. Then V is

Let V be the volume of the solid generated by revolving the region enclosed by y = x³, y = 2x² - x³; 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 = 2. The cross-sections of this solid perpendicular to the x-axis run from y = 0 to y = x² 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 = 3. The cross-sections of this solid perpendicular to the x-axis run from $y = x ^ { 2 }$ to $y = 3 \sqrt { 3 x }$ and they are squares with bases in the xy-plane. Then V is

The work done by a variable force $F ( x ) = x ^ { 2 } + 4$ Newtons that moves an object along a straight line in the direction of F from x = 1 meters to x = 3 meters, in Newton-meters, is

Let V be the volume of a wedge cut from a right-circular cylinder with a radius of 1 inch by two planes, one perpendicular to the axis of the cylinder and the other intersecting the first along a diameter of the circular plane section at an angle of $\frac { \pi } { 3 }$ radian. Then V is

A trough whose cross-section is a trapezoid whose lower base is 4 feet long and upper base is 8 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

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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 y-axis, y = 4; about y = 4. 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 xˉ\bar { x }xˉ denote the x-coordinate of the centroid of the region enclosed by y=x2y = x ^ { 2 }y=x2 and y=2x+3y = 2 x + 3y=2x+3 Then xˉ\bar { x }xˉ =

By the Pappus Theorem, the volume of the solid formed by revolving the region enclosed by the circle (x−4)2+y2=9( x - 4 ) ^ { 2 } + y ^ { 2 } = 9(x−4)2+y2=9 about the y-axis 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 trough cross-section is a 6-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 V be the volume of the solid generated by revolving the region enclosed by about the x-axis. Then V is

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

The arc length of y=x36+12x for x∈[1,4]y = \frac { x ^ { 3 } } { 6 } + \frac { 1 } { 2 x } \text { for } x \in [ 1,4 ]y=6x3​+2x1​ for x∈[1,4] is

Let A denote the area enclosed by the equations x=y3−2yx = y ^ { 3 } - 2 yx=y3−2y and x=7y.x = 7 y .x=7y. Then A is

The work done by a winch winding in a 50-meter chain weighing 3 Newtons per meter, in Newton-meters, is ​

Let V be the volume of the solid generated by revolving the region enclosed by about x = 2. Then V is

Let V be the volume of the solid generated by revolving the region enclosed by y = x3, y = 2x2 - x3; 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 = 2. The cross-sections of this solid perpendicular to the x-axis run from y = 0 to y = x2 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 = 3. The cross-sections of this solid perpendicular to the x-axis run from y=x2y = x ^ { 2 }y=x2 to y=33xy = 3 \sqrt { 3 x }y=33x​ and they are squares with bases in the xy-plane. Then V is

The work done by a variable force F(x)=x2+4F ( x ) = x ^ { 2 } + 4F(x)=x2+4 Newtons that moves an object along a straight line in the direction of F from x = 1 meters to x = 3 meters, in Newton-meters, is

Let V be the volume of a wedge cut from a right-circular cylinder with a radius of 1 inch by two planes, one perpendicular to the axis of the cylinder and the other intersecting the first along a diameter of the circular plane section at an angle of π3\frac { \pi } { 3 }3π​ radian. Then V is

A trough whose cross-section is a trapezoid whose lower base is 4 feet long and upper base is 8 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 $y = \sqrt { x },$ the y-axis, y = 4; about y = 4. 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 $\bar { x }$ denote the x-coordinate of the centroid of the region enclosed by $y = x ^ { 2 }$ and $y = 2 x + 3$ Then $\bar { x }$ =

By the Pappus Theorem, the volume of the solid formed by revolving the region enclosed by the circle $( x - 4 ) ^ { 2 } + y ^ { 2 } = 9$ about the y-axis 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

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

The arc length of $y = \frac { x ^ { 3 } } { 6 } + \frac { 1 } { 2 x } \text { for } x \in [ 1,4 ]$ is

Let A denote the area enclosed by the equations $x = y ^ { 3 } - 2 y$ and $x = 7 y .$ Then A is

The work done by a winch winding in a 50-meter chain weighing 3 Newtons per meter, in Newton-meters, is

Let V be the volume of the solid generated by revolving the region enclosed by y = x³, y = 2x² - x³; 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 = 2. The cross-sections of this solid perpendicular to the x-axis run from y = 0 to y = x² 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 = 3. The cross-sections of this solid perpendicular to the x-axis run from $y = x ^ { 2 }$ to $y = 3 \sqrt { 3 x }$ and they are squares with bases in the xy-plane. Then V is

The work done by a variable force $F ( x ) = x ^ { 2 } + 4$ Newtons that moves an object along a straight line in the direction of F from x = 1 meters to x = 3 meters, in Newton-meters, is

Let V be the volume of a wedge cut from a right-circular cylinder with a radius of 1 inch by two planes, one perpendicular to the axis of the cylinder and the other intersecting the first along a diameter of the circular plane section at an angle of $\frac { \pi } { 3 }$ radian. Then V is