Question 1

A heavy rope, 40 ft long, weighs $0.8$ lb/ft and hangs over the edge of a building 110 ft high. How much work is done in pulling the rope to the top of the building?

Accepted Answer

A)  $710 \mathrm{ft}-\mathrm{lb}$ 
B)  $730 \mathrm{ft}-\mathrm{lb}$ 
C)  $750 \mathrm{ft}-\mathrm{lb}$ 
D)  $740 \mathrm{ft}-\mathrm{lb}$ 
E)  $640 \mathrm{ft}-\mathrm{lb}$ 
A)  $710 \mathrm{ft}-\mathrm{lb}$ 
B)  $730 \mathrm{ft}-\mathrm{lb}$ 
C)  $750 \mathrm{ft}-\mathrm{lb}$ 
D)  $740 \mathrm{ft}-\mathrm{lb}$ 
E)  $640 \mathrm{ft}-\mathrm{lb}$

Question 2

Use the Midpoint Rule with n = 4 to estimate the volume obtained by rotating about the region under the y-axis the region under the curve. $y=\tan x, \quad 0 \leq x \leq \frac{\pi}{4}$ Select the correct answer. The choices are rounded to the nearest hundredth.

Accepted Answer

A)  $V=0.156$ 
B)  $V=1.851$ 
C)  $V=0.825$ 
D)  $V=1.142$ 
E)  $V=0.491$ 
A)  $V=0.156$ 
B)  $V=1.851$ 
C)  $V=0.825$ 
D)  $V=1.142$ 
E)  $V=0.491$

Question 3

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle. $y=\frac{1}{x\left(x^{2}+144\right)}$ , y = 0, x = 1, x = 12; the y-axis

Accepted Answer

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

Question 4

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 = $\sqrt{x-1}$ , y = x - 1; the line x = 2

Accepted Answer

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

Question 5

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. Round your answer to 3 decimal places. $y=x^{2}+3 x-10$ about the y-axis and $y=0$ .

Accepted Answer

A)  1763.213 
B)  None of these 
C)  1752.016 
D)  1760.025 
E)  880.012 
A)  1763.213 
B)  None of these 
C)  1752.016 
D)  1760.025 
E)  880.012

Question 6

A spring has a natural length of 20 cm. If a force of 25 N is required to keep it stretched to a length of 30 cm, how much work is required to stretch it from 20 cm to 32 cm?

Accepted Answer

A)  $1.80 \mathrm{~J}$ 
B)  $3.80 \mathrm{~J}$ 
C)  $5.80 \mathrm{~J}$ 
D)  $4.80 \mathrm{~J}$ 
E)  $2.80 \mathrm{~J}$ 
A)  $1.80 \mathrm{~J}$ 
B)  $3.80 \mathrm{~J}$ 
C)  $5.80 \mathrm{~J}$ 
D)  $4.80 \mathrm{~J}$ 
E)  $2.80 \mathrm{~J}$

Question 7

Sketch the region bounded by the graphs of the given equations and find the area of their region. $y=\frac{x}{x^{2}+5}$ , $y=-\frac{1}{6} x^{2}$ , $x=1$

Accepted Answer

A)    $\ln \left(\frac{6}{5}\right)$ 
B)    $4-\ln \left(\frac{6}{5}\right)$ 
C)    $\frac{1}{2} \ln \left(\frac{6}{5}\right)+\frac{1}{18}$ 
D)    $\frac{1}{2} \ln \left(\frac{6}{5}\right)-\frac{1}{18}$ 
A)    $\ln \left(\frac{6}{5}\right)$ 
B)    $4-\ln \left(\frac{6}{5}\right)$ 
C)    $\frac{1}{2} \ln \left(\frac{6}{5}\right)+\frac{1}{18}$ 
D)    $\frac{1}{2} \ln \left(\frac{6}{5}\right)-\frac{1}{18}$

Question 8

Sketch the region enclosed by $x=1-y^{2} \text { and } x=y^{2}-1$ Find the area of the region.

Accepted Answer

The answer of Sketch the region enclosed by \(x=1-y^{2} \text...

Question 9

Find the number(s) a such that the average value of the function $f(x)=50-28 x+3 x^{2}$ on the interval $[0, a]$ is equal to 10.

Accepted Answer

A)  $a=10$ 
B)  $a=-4$ 
C)  $a=-10$ 
D)  $a=4$ 
E)  $a=-10$. 
A)  $a=10$ 
B)  $a=-4$ 
C)  $a=-10$ 
D)  $a=4$ 
E)  $a=-10$.

Question 10

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y = cos x + 1, x = 0, y = 0, x = $\frac{1}{2}$ $\pi$

Accepted Answer

A)  $\frac{3}{4}$  $\pi^{2}$ + 2$\pi$ 
B)  $\frac{3}{4}$ $\pi^{2}$ + 4$\pi$ 
C) 
$\pi^{2}$ + 4$\pi$ 
D) 
$\pi^{2}$  +2$\pi$ 
A)  $\frac{3}{4}$  $\pi^{2}$ + 2$\pi$ 
B)  $\frac{3}{4}$ $\pi^{2}$ + 4$\pi$ 
C) 
$\pi^{2}$ + 4$\pi$ 
D) 
$\pi^{2}$  +2$\pi$

Question 11

Find the volume of the solid that is obtained by revolving the region about the line y = $\frac{5}{2}$ .

Accepted Answer

A)  $\frac{89}{42}$  $\pi$ 
B)  $\frac{89}{14}$ $\pi$ 
C)  $\frac{89}{84}$  $\pi$ 
D)  89 $\pi$ 
A)  $\frac{89}{42}$  $\pi$ 
B)  $\frac{89}{14}$ $\pi$ 
C)  $\frac{89}{84}$  $\pi$ 
D)  89 $\pi$

Question 12

Find the area of the region bounded by the parabola $y=x^{2}$ , the tangent line to this parabola at $(1,1)$ , and the x axis.

Accepted Answer

A)  $1 \frac{2}{3}$ 
B)  $\frac{1}{12}$ 
C)  $\frac{2}{3}$ 
D)  $1 \frac{1}{3}$ 
E)  $47 \frac{2}{3}$ 
A)  $1 \frac{2}{3}$ 
B)  $\frac{1}{12}$ 
C)  $\frac{2}{3}$ 
D)  $1 \frac{1}{3}$ 
E)  $47 \frac{2}{3}$

Question 13

Graph the region between the curves and use your calculator to compute the area correct to five decimal places. $y=5 e^{1-x^{2}}, y=5 x^{4}$

Accepted Answer

A)  12.08127 
B)  91.504 
C)  3.141257 
D)  3.66016 
E)  18.3008 
A)  12.08127 
B)  91.504 
C)  3.141257 
D)  3.66016 
E)  18.3008

Question 14

Find the volume of the solid obtained by rotating the region bounded by $y=2 x^{3}$ and $x=y^{3}$ about the line $x=-2 \text {. }$

Accepted Answer

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

Question 15

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle.
y = 3 $x^{2}$ , y = 0, x = 1; the y-axis

Accepted Answer

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

A heavy rope, 40 ft long, weighs $0.8$ lb/ft and hangs over the edge of a building 110 ft high. How much work is done in pulling the rope to the top of the building?

Use the Midpoint Rule with n = 4 to estimate the volume obtained by rotating about the region under the y-axis the region under the curve. $y=\tan x, \quad 0 \leq x \leq \frac{\pi}{4}$ Select the correct answer. The choices are rounded to the nearest hundredth.

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle. $y=\frac{1}{x\left(x^{2}+144\right)}$ , y = 0, x = 1, x = 12; the y-axis

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 = $\sqrt{x-1}$ , y = x - 1; the line x = 2

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. Round your answer to 3 decimal places. $y=x^{2}+3 x-10$ about the y-axis and $y=0$ .

A spring has a natural length of 20 cm. If a force of 25 N is required to keep it stretched to a length of 30 cm, how much work is required to stretch it from 20 cm to 32 cm?

Sketch the region bounded by the graphs of the given equations and find the area of their region. $y=\frac{x}{x^{2}+5}$ , $y=-\frac{1}{6} x^{2}$ , $x=1$

Sketch the region enclosed by $x=1-y^{2} \text { and } x=y^{2}-1$ Find the area of the region.

Find the number(s) a such that the average value of the function $f(x)=50-28 x+3 x^{2}$ on the interval $[0, a]$ is equal to 10.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y = cos x + 1, x = 0, y = 0, x = $\frac{1}{2}$ $\pi$

Find the volume of the solid that is obtained by revolving the region about the line y = $\frac{5}{2}$ . $Find the volume of the solid that is obtained by revolving the region about the line y = \frac{5}{2} .$

Find the area of the region bounded by the parabola $y=x^{2}$ , the tangent line to this parabola at $(1,1)$ , and the x axis.

Graph the region between the curves and use your calculator to compute the area correct to five decimal places. $y=5 e^{1-x^{2}}, y=5 x^{4}$

Find the volume of the solid obtained by rotating the region bounded by $y=2 x^{3}$ and $x=y^{3}$ about the line $x=-2 \text {. }$

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle. y = 3 $x^{2}$ , y = 0, x = 1; the y-axis

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

A heavy rope, 40 ft long, weighs 0.80.80.8 lb/ft and hangs over the edge of a building 110 ft high. How much work is done in pulling the rope to the top of the building?

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 = x−1\sqrt{x-1}x−1​ , y = x - 1; the line x = 2

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. Round your answer to 3 decimal places. y=x2+3x−10y=x^{2}+3 x-10y=x2+3x−10 about the y-axis and y=0y=0y=0 .

A spring has a natural length of 20 cm. If a force of 25 N is required to keep it stretched to a length of 30 cm, how much work is required to stretch it from 20 cm to 32 cm?

Sketch the region bounded by the graphs of the given equations and find the area of their region. y=xx2+5y=\frac{x}{x^{2}+5}y=x2+5x​ , y=−16x2y=-\frac{1}{6} x^{2}y=−61​x2 , x=1x=1x=1

Sketch the region enclosed by x=1−y2 and x=y2−1x=1-y^{2} \text { and } x=y^{2}-1x=1−y2 and x=y2−1 Find the area of the region.

Find the number(s) a such that the average value of the function f(x)=50−28x+3x2f(x)=50-28 x+3 x^{2}f(x)=50−28x+3x2 on the interval [0,a][0, a][0,a] is equal to 10.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y = cos x + 1, x = 0, y = 0, x = 12\frac{1}{2}21​ π\piπ

Find the volume of the solid that is obtained by revolving the region about the line y = 52\frac{5}{2}25​ .

Find the area of the region bounded by the parabola y=x2y=x^{2}y=x2 , the tangent line to this parabola at (1,1)(1,1)(1,1) , and the x axis.

Graph the region between the curves and use your calculator to compute the area correct to five decimal places. y=5e1−x2,y=5x4y=5 e^{1-x^{2}}, y=5 x^{4}y=5e1−x2,y=5x4

Find the volume of the solid obtained by rotating the region bounded by y=2x3y=2 x^{3}y=2x3 and x=y3x=y^{3}x=y3 about the line x=−2. x=-2 \text {. }x=−2.

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle. y = 3 x2x^{2}x2 , y = 0, x = 1; the y-axis

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

A heavy rope, 40 ft long, weighs $0.8$ lb/ft and hangs over the edge of a building 110 ft high. How much work is done in pulling the rope to the top of the building?

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 = $\sqrt{x-1}$ , y = x - 1; the line x = 2

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. Round your answer to 3 decimal places. $y=x^{2}+3 x-10$ about the y-axis and $y=0$ .

Sketch the region bounded by the graphs of the given equations and find the area of their region. $y=\frac{x}{x^{2}+5}$ , $y=-\frac{1}{6} x^{2}$ , $x=1$

Sketch the region enclosed by $x=1-y^{2} \text { and } x=y^{2}-1$ Find the area of the region.

Find the number(s) a such that the average value of the function $f(x)=50-28 x+3 x^{2}$ on the interval $[0, a]$ is equal to 10.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y = cos x + 1, x = 0, y = 0, x = $\frac{1}{2}$ $\pi$

Find the volume of the solid that is obtained by revolving the region about the line y = $\frac{5}{2}$ . $Find the volume of the solid that is obtained by revolving the region about the line y = \frac{5}{2} .$

Find the area of the region bounded by the parabola $y=x^{2}$ , the tangent line to this parabola at $(1,1)$ , and the x axis.

Graph the region between the curves and use your calculator to compute the area correct to five decimal places. $y=5 e^{1-x^{2}}, y=5 x^{4}$

Find the volume of the solid obtained by rotating the region bounded by $y=2 x^{3}$ and $x=y^{3}$ about the line $x=-2 \text {. }$

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle. y = 3 $x^{2}$ , y = 0, x = 1; the y-axis