Question 1

Find the volume of the solid obtained by revolving the region under the graph of $y=5\left(\frac{1}{x^{2}}-\frac{1}{x^{4}}\right)$ on [1, $\infty$ ) about the x-axis.

Accepted Answer

The answer of Find the volume of the solid obtained...

Question 2

Find the area of the region bounded by the curves. $x=2 y^{2}, x=16+y^{2}$

Accepted Answer

The answer of Find the area of the region bounded...

Question 3

If 10 J of work are needed to stretch a spring from 10 cm to 12 cm and another 20 J are needed to stretch it from 12 cm to 14 cm, what is the natural length of the spring? Round the answer to nearest integer.

Accepted Answer

A)  $10 \text { cm }$ 
B)  $11 \text { cm }$ 
C)  $6 \mathrm{~cm}$ 
D)  $8 \mathrm{~cm}$ 
E)  $9 \mathrm{~cm}$ 
A)  $10 \text { cm }$ 
B)  $11 \text { cm }$ 
C)  $6 \mathrm{~cm}$ 
D)  $8 \mathrm{~cm}$ 
E)  $9 \mathrm{~cm}$

Question 4

Use a graphing utility to (a) plot the graphs of the given functions and (b) find the x-coordinates of the points of intersection of the curves. Then find an approximation of the area of the region bounded by the curves using the integration capabilities of the graphing utility. Round answers to two decimal places.
y = 3 $x^{2}$ , y = 5 - $x^{4}$

Accepted Answer

The answer of Use a graphing utility to (a) plot...

Question 5

Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the x-axis. $y^{2}-3 y+x=0, x=0$

Accepted Answer

The answer of Use the method of cylindrical shells to...

Question 6

Sketch the region bounded by the graphs of the given equations and find the area of their region. $x=y^{2}+3$ $x=y-2$ , $y=-1$ , $y=1$

Accepted Answer

A)    $-\frac{16}{3}$ 
B)    $-\frac{32}{3}$ 
C)    $\frac{32}{3}$ 
D)    16 
A)    $-\frac{16}{3}$ 
B)    $-\frac{32}{3}$ 
C)    $\frac{32}{3}$ 
D)    16

Question 7

Find the volume of the solid obtained by rotating the region bounded by $y=2 \sqrt[4]{x} \text { and } y=2 x$ about the line $y=2 .$

Accepted Answer

A)  $\frac{16 \pi}{15}$ 
B)  $\frac{2}{9}$ 
C)  $\frac{1}{16}$ 
D)  $\frac{\pi}{9}$ 
E)  $\frac{8 \pi}{9}$ 
A)  $\frac{16 \pi}{15}$ 
B)  $\frac{2}{9}$ 
C)  $\frac{1}{16}$ 
D)  $\frac{\pi}{9}$ 
E)  $\frac{8 \pi}{9}$

Question 8

Sketch the region bounded by the graphs of the given equations and find the area of that region.
y = sin 2x, y = 7cos x, x = $\frac{\pi}{2}$ , x = $\frac{3 \pi}{2}$

Accepted Answer

The answer of Sketch the region bounded by the graphs...

Question 9

A particle moves a distance of 150 ft along a straight line. As it moves, it is acted upon by a constant force of magnitude 15 lb in a direction opposite to that of the motion. What is the work done by the force?

Accepted Answer

A)  $-\frac{1}{2250}$ ft-lb 
B)  $-10$ ft-lb 
C)  -2250 ft-lb 
D)  $-\frac{1}{10}$ ft-lb 
A)  $-\frac{1}{2250}$ ft-lb 
B)  $-10$ ft-lb 
C)  -2250 ft-lb 
D)  $-\frac{1}{10}$ ft-lb

Question 10

Find the volume of the solid obtained by rotating the region bounded by $y=x^{3} \text { and } x=y^{3}$ about the x-axis.

Accepted Answer

A)  $\frac{16}{7} \pi$ 
B)  $\frac{16}{35} \pi$ 
C)  $\frac{7}{2}$ 
D)  $\frac{18}{35}$ 
E)  $16 \pi$ 
A)  $\frac{16}{7} \pi$ 
B)  $\frac{16}{35} \pi$ 
C)  $\frac{7}{2}$ 
D)  $\frac{18}{35}$ 
E)  $16 \pi$

Question 11

If a force of 6 lbs is required to hold a spring stretched 5 inches beyond its natural length, then 38.4 lb-in. of work is done in stretching it from its natural length to 8 in. beyond its natural length.

Accepted Answer

A) True 
 B)False

Question 12

Sketch the region bounded by the graphs of the given equations and find the area of that region. $y=x^{2}-2 x-1$ , $y=-e^{x}-1$ , $x=0$ , $x=1$

Accepted Answer

The answer of Sketch the region bounded by the graphs...

Question 13

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis.
y = $x^{2}$ , y = 0, x = 3, x = 5; the x-axis

Accepted Answer

The answer of Find the volume of the solid generated...

Question 14

Sketch a plane region and indicate the axis about which it is revolved so that the resulting solid of revolution (found using the shell method) is given by the integral. 2$\pi$ $\int_{0}^{1} y\left(y^{1 / 9}-y\right) d y$

Accepted Answer

A)    x-axis 
B)    y-axis 
C)    x-axis 
D)    y-axis 
A)    x-axis 
B)    y-axis 
C)    x-axis 
D)    y-axis

Question 15

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. $y=5\left(x^{2}+4\right), y=5\left(12-x^{2}\right) ; \text { about } y=-5$

Accepted Answer

A)  $9550 \pi$ 
B)  $9525 \pi$ 
C)  $9575 \pi$ 
D)  $9600 \pi$ 
E)  None of these 
A)  $9550 \pi$ 
B)  $9525 \pi$ 
C)  $9575 \pi$ 
D)  $9600 \pi$ 
E)  None of these

Question 16

The base of S is a circular region with boundary curve $6 x^{2}+6 y^{2}=24$ Cross-sections perpendicular to the x axis are isosceles right triangles with hypotenuse in the base.
Find the volume of S.

Accepted Answer

The answer of The base of S is a circular...

Question 17

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. Sketch the region and a representative rectangle. y = 25 - $x^{2}$ , y = 0; the line x = - 5

Accepted Answer

A)  $\frac{5000}{3}$ $\pi$  
B)  $1250$ $\pi$  
C)  $\frac{2500}{3}$ $\pi$  
D)  $625$ $\pi$  
A)  $\frac{5000}{3}$ $\pi$  
B)  $1250$ $\pi$  
C)  $\frac{2500}{3}$ $\pi$  
D)  $625$ $\pi$

Question 18

Find the area of the shaded region.

Accepted Answer

A)  $\frac{112}{3}$ 
B)  $\frac{56}{3}$ 
C)  $\frac{112}{9}$ 
D)  $\frac{224}{3}$ 
A)  $\frac{112}{3}$ 
B)  $\frac{56}{3}$ 
C)  $\frac{112}{9}$ 
D)  $\frac{224}{3}$

Question 19

Use integration to find the area of the triangle with the given vertices. (0, 0), (1, 7), (4, 2)

Accepted Answer

A)  $\frac{13}{2}$ 
B)  26 
C)  28 
D)  13 
A)  $\frac{13}{2}$ 
B)  26 
C)  28 
D)  13

Question 20

Find the area of the region bounded by the given curves. $y=x^{2}-6 x, y=5 x+12$

Accepted Answer

A)  $2197$ 
B)  $\frac{25}{3}$ 
C)  $\frac{2197}{3}$ 
D)  $\frac{2197}{6}$ 
E)  6 
A)  $2197$ 
B)  $\frac{25}{3}$ 
C)  $\frac{2197}{3}$ 
D)  $\frac{2197}{6}$ 
E)  6

Find the volume of the solid obtained by revolving the region under the graph of $y=5\left(\frac{1}{x^{2}}-\frac{1}{x^{4}}\right)$ on [1, $\infty$ ) about the x-axis.

Find the area of the region bounded by the curves. $x=2 y^{2}, x=16+y^{2}$

If 10 J of work are needed to stretch a spring from 10 cm to 12 cm and another 20 J are needed to stretch it from 12 cm to 14 cm, what is the natural length of the spring? Round the answer to nearest integer.

Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the x-axis. $y^{2}-3 y+x=0, x=0$

Sketch the region bounded by the graphs of the given equations and find the area of their region. $x=y^{2}+3$ $x=y-2$ , $y=-1$ , $y=1$

Find the volume of the solid obtained by rotating the region bounded by $y=2 \sqrt[4]{x} \text { and } y=2 x$ about the line $y=2 .$

Sketch the region bounded by the graphs of the given equations and find the area of that region. y = sin 2x, y = 7cos x, x = $\frac{\pi}{2}$ , x = $\frac{3 \pi}{2}$

A particle moves a distance of 150 ft along a straight line. As it moves, it is acted upon by a constant force of magnitude 15 lb in a direction opposite to that of the motion. What is the work done by the force?

Find the volume of the solid obtained by rotating the region bounded by $y=x^{3} \text { and } x=y^{3}$ about the x-axis.

If a force of 6 lbs is required to hold a spring stretched 5 inches beyond its natural length, then 38.4 lb-in. of work is done in stretching it from its natural length to 8 in. beyond its natural length.

Sketch the region bounded by the graphs of the given equations and find the area of that region. $y=x^{2}-2 x-1$ , $y=-e^{x}-1$ , $x=0$ , $x=1$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. y = $x^{2}$ , y = 0, x = 3, x = 5; the x-axis

Sketch a plane region and indicate the axis about which it is revolved so that the resulting solid of revolution (found using the shell method) is given by the integral. 2 $\pi$ $\int_{0}^{1} y\left(y^{1 / 9}-y\right) d y$

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. $y=5\left(x^{2}+4\right), y=5\left(12-x^{2}\right) ; \text { about } y=-5$

The base of S is a circular region with boundary curve $6 x^{2}+6 y^{2}=24$ Cross-sections perpendicular to the x axis are isosceles right triangles with hypotenuse in the base. Find the volume of S.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. Sketch the region and a representative rectangle. y = 25 - $x^{2}$ , y = 0; the line x = - 5

Find the area of the shaded region.

Use integration to find the area of the triangle with the given vertices. (0, 0), (1, 7), (4, 2)

Find the area of the region bounded by the given curves. $y=x^{2}-6 x, y=5 x+12$

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Exam 6: Inverse Functions

Find the volume of the solid obtained by revolving the region under the graph of y=5(1x2−1x4)y=5\left(\frac{1}{x^{2}}-\frac{1}{x^{4}}\right)y=5(x21​−x41​) on [1, ∞\infty∞ ) about the x-axis.

Find the area of the region bounded by the curves. x=2y2,x=16+y2x=2 y^{2}, x=16+y^{2}x=2y2,x=16+y2

If 10 J of work are needed to stretch a spring from 10 cm to 12 cm and another 20 J are needed to stretch it from 12 cm to 14 cm, what is the natural length of the spring? Round the answer to nearest integer.

Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the x-axis. y2−3y+x=0,x=0y^{2}-3 y+x=0, x=0y2−3y+x=0,x=0

Sketch the region bounded by the graphs of the given equations and find the area of their region. x=y2+3x=y^{2}+3x=y2+3 x=y−2x=y-2x=y−2 , y=−1y=-1y=−1 , y=1y=1y=1

Find the volume of the solid obtained by rotating the region bounded by y=2x4 and y=2xy=2 \sqrt[4]{x} \text { and } y=2 xy=24x​ and y=2x about the line y=2.y=2 .y=2.

Sketch the region bounded by the graphs of the given equations and find the area of that region. y = sin 2x, y = 7cos x, x = π2\frac{\pi}{2}2π​ , x = 3π2\frac{3 \pi}{2}23π​

A particle moves a distance of 150 ft along a straight line. As it moves, it is acted upon by a constant force of magnitude 15 lb in a direction opposite to that of the motion. What is the work done by the force?

Find the volume of the solid obtained by rotating the region bounded by y=x3 and x=y3y=x^{3} \text { and } x=y^{3}y=x3 and x=y3 about the x-axis.

If a force of 6 lbs is required to hold a spring stretched 5 inches beyond its natural length, then 38.4 lb-in. of work is done in stretching it from its natural length to 8 in. beyond its natural length.

Sketch the region bounded by the graphs of the given equations and find the area of that region. y=x2−2x−1y=x^{2}-2 x-1y=x2−2x−1 , y=−ex−1y=-e^{x}-1y=−ex−1 , x=0x=0x=0 , x=1x=1x=1

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. y = x2x^{2}x2 , y = 0, x = 3, x = 5; the x-axis

Sketch a plane region and indicate the axis about which it is revolved so that the resulting solid of revolution (found using the shell method) is given by the integral. 2 π\piπ ∫01y(y1/9−y)dy\int_{0}^{1} y\left(y^{1 / 9}-y\right) d y∫01​y(y1/9−y)dy

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y=5(x2+4),y=5(12−x2); about y=−5y=5\left(x^{2}+4\right), y=5\left(12-x^{2}\right) ; \text { about } y=-5y=5(x2+4),y=5(12−x2); about y=−5

The base of S is a circular region with boundary curve 6x2+6y2=246 x^{2}+6 y^{2}=246x2+6y2=24 Cross-sections perpendicular to the x axis are isosceles right triangles with hypotenuse in the base. Find the volume of S.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. Sketch the region and a representative rectangle. y = 25 - x2x^{2}x2 , y = 0; the line x = - 5

Find the area of the shaded region.

Use integration to find the area of the triangle with the given vertices. (0, 0), (1, 7), (4, 2)

Find the area of the region bounded by the given curves. y=x2−6x,y=5x+12y=x^{2}-6 x, y=5 x+12y=x2−6x,y=5x+12

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Find the volume of the solid obtained by revolving the region under the graph of $y=5\left(\frac{1}{x^{2}}-\frac{1}{x^{4}}\right)$ on [1, $\infty$ ) about the x-axis.

Find the area of the region bounded by the curves. $x=2 y^{2}, x=16+y^{2}$

Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the x-axis. $y^{2}-3 y+x=0, x=0$

Sketch the region bounded by the graphs of the given equations and find the area of their region. $x=y^{2}+3$ $x=y-2$ , $y=-1$ , $y=1$

Find the volume of the solid obtained by rotating the region bounded by $y=2 \sqrt[4]{x} \text { and } y=2 x$ about the line $y=2 .$

Sketch the region bounded by the graphs of the given equations and find the area of that region. y = sin 2x, y = 7cos x, x = $\frac{\pi}{2}$ , x = $\frac{3 \pi}{2}$

Find the volume of the solid obtained by rotating the region bounded by $y=x^{3} \text { and } x=y^{3}$ about the x-axis.

Sketch the region bounded by the graphs of the given equations and find the area of that region. $y=x^{2}-2 x-1$ , $y=-e^{x}-1$ , $x=0$ , $x=1$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. y = $x^{2}$ , y = 0, x = 3, x = 5; the x-axis

Sketch a plane region and indicate the axis about which it is revolved so that the resulting solid of revolution (found using the shell method) is given by the integral. 2 $\pi$ $\int_{0}^{1} y\left(y^{1 / 9}-y\right) d y$

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. $y=5\left(x^{2}+4\right), y=5\left(12-x^{2}\right) ; \text { about } y=-5$

The base of S is a circular region with boundary curve $6 x^{2}+6 y^{2}=24$ Cross-sections perpendicular to the x axis are isosceles right triangles with hypotenuse in the base. Find the volume of S.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. Sketch the region and a representative rectangle. y = 25 - $x^{2}$ , y = 0; the line x = - 5

Find the area of the region bounded by the given curves. $y=x^{2}-6 x, y=5 x+12$