Question 1

Find the values of c such that the area of the region bounded by the parabolas $y=x^{2}-c^{2} \text { and } y=c^{2}-x^{2}$ is $27$ .

Accepted Answer

$\pm 3$

Question 2

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. $x^{2}+(y-1)^{2}=1^{2}$ about the y-axis

Accepted Answer

A)  $V=\frac{4}{3} \pi$ 
B)  $V=\frac{8}{3} \pi$ 
C)  $V=\frac{28}{3} \pi$ 
D)  $V=\frac{1}{3} \pi$ 
E)  $V=\frac{20}{3} \pi$ 
A)  $V=\frac{4}{3} \pi$ 
B)  $V=\frac{8}{3} \pi$ 
C)  $V=\frac{28}{3} \pi$ 
D)  $V=\frac{1}{3} \pi$ 
E)  $V=\frac{20}{3} \pi$ A

Question 3

Racing cars driven by Chris and Kelly are side by side at the start of a race. The table shows the velocities of each car (in miles per hour) during the first ten seconds of the race. Use the Midpoint Rule to estimate how much farther Kelly travels than Chris does during the first ten seconds. $\begin{array}{|c|c|c|}
\hline t & V_{c} & V_{k} \
\hline 0 & 0 & 0 \
\hline 1 & 22 & 28 \
\hline 2 & 33 & 38 \
\hline 3 & 45 & 46 \
\hline 4 & 53 & 60 \
\hline 5 & 63 & 69 \
\hline 6 & 72 & 83 \
\hline 7 & 78 & 83 \
\hline 8 & 85 & 97 \
\hline 9 & 90 & 98 \
\hline 10 & 90 & 102 \
\hline
\end{array}$

Accepted Answer

$\frac{1144}{15}$ ft

Question 4

Find the work done in pushing a car a distance of 14 m while exerting a constant force of 370 N.

Accepted Answer

The answer of Find the work done in pushing a...

Question 5

Find the area of the region bounded by the given curves. $y=\cos x, y=\sin 3 x, x=0, x=\frac{\pi}{2}$

Accepted Answer

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

Question 6

Find the volume common to two spheres, each with radius r = $12$ if the center of each sphere lies on the surface of the other sphere.

Accepted Answer

The answer of Find the volume common to two spheres,...

Question 7

The base of S is the parabolic region $\left\{(x, y) \mid x^{2} \leq y \leq 4\right\} .$ Cross-sections perpendicular to the y axis are squares.
Find the volume of S.

Accepted Answer

The answer of The base of S is the parabolic...

Question 8

Use the method of disks or washers, or 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. $y=2 \sqrt{x}$ , $y=x-3$ , $y=0$ , the x-axis

Accepted Answer

A)  $45 \pi$   
B)  $135 \pi$   
C)  $180 \pi$   
D)  $90 \pi$   
A)  $45 \pi$   
B)  $135 \pi$   
C)  $180 \pi$   
D)  $90 \pi$

Question 9

Find the volume of a cap of a sphere with radius r = $100$ and height h = 3 9 .

Accepted Answer

The answer of Find the volume of a cap of...

Question 10

Use a graphing utility to (a) plot the graphs of the given functions, (b) find the approximate x-coordinates of the points of intersection of the graphs, and (c) find an approximation of the volume of the solid obtained by revolving the region bounded by the graphs of the functions about the y-axis. Round answers to two decimal places.
y = x, y = $x^{5}$ - $x^{2}$ , x $\ge$ 0

Accepted Answer

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

Question 11

Use the method of disks or washers, or 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 = $x^{2}$ , y = 2x - 1, y = 4; the y-axis

Accepted Answer

The answer of Use the method of disks or washers,...

Question 12

Find the average value of the function $z(t)=3 \cos (t)$ on the interval $\left[0, \frac{\pi}{2}\right]$ .

Accepted Answer

The answer of Find the average value of the function...

Question 13

Find the volume of a pyramid with height 4 and base an equilateral triangle with side a = 4 .

Accepted Answer

A)  $\frac{16 \sqrt{3}}{3}$ 
B)  $\frac{3 \sqrt{3}}{2}$ 
C)  $16 \sqrt{3}$ 
D)  $\frac{16 \sqrt{3}}{5}$ 
E)  $\frac{\sqrt{3}}{10}$ 
A)  $\frac{16 \sqrt{3}}{3}$ 
B)  $\frac{3 \sqrt{3}}{2}$ 
C)  $16 \sqrt{3}$ 
D)  $\frac{16 \sqrt{3}}{5}$ 
E)  $\frac{\sqrt{3}}{10}$

Question 14

Find the area of the region bounded by the curves. $y=\cos x, y=\sin 3 x, x=0, x=\frac{\pi}{2}$

Accepted Answer

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

Question 15

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. $y=3\left(x^{2}+4\right), y=3\left(12-x^{2}\right) ; \text { about } y=-3$

Accepted Answer

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

Question 16

Find the average value of the function $f(t)=9 t \sin \left(t^{2}\right)$ on the interval $[0,20]$ . Round your answer to 3 decimal places.

Accepted Answer

A)  9.342 
B)  0.45 
C)  0.432 
D)  $0.342$ 
E)  18 
A)  9.342 
B)  0.45 
C)  0.432 
D)  $0.342$ 
E)  18

Question 17

The height of a monument is $20$ m. A horizontal cross-section at a distance x meters from the top is an equilateral triangle with side $\frac{x}{4}$ meters. Find the volume of the monument.

Accepted Answer

A)  $\frac{125 \sqrt{3}}{3} \mathrm{~m}^{3}$ 
B)  $\frac{125 \sqrt{2}}{3} \mathrm{~m}^{3}$ 
C)  $\frac{\sqrt{3}}{3} \mathrm{~m}^{3}$ 
D)  $\frac{125 \sqrt{2}}{2} \mathrm{~m}^{3}$ 
E)  $125 \sqrt{3} \mathrm{~m}^{3}$ 
A)  $\frac{125 \sqrt{3}}{3} \mathrm{~m}^{3}$ 
B)  $\frac{125 \sqrt{2}}{3} \mathrm{~m}^{3}$ 
C)  $\frac{\sqrt{3}}{3} \mathrm{~m}^{3}$ 
D)  $\frac{125 \sqrt{2}}{2} \mathrm{~m}^{3}$ 
E)  $125 \sqrt{3} \mathrm{~m}^{3}$

Question 18

Find the volume of the solid obtained by rotating about the x-axis the region under the curve $y=\frac{1}{x}$ from x = 7 to x = 8 .

Accepted Answer

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

Question 19

Find the area of the shaded region.

Accepted Answer

A)  7 
B)  $\frac{7}{12}$ 
C)  $\frac{1}{6}$ 
D)  $\frac{7}{6}$ 
A)  7 
B)  $\frac{7}{12}$ 
C)  $\frac{1}{6}$ 
D)  $\frac{7}{6}$

Question 20

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 = $\frac{1}{x}$ , y = 0, x = 2, x = 5; the y-axis

Accepted Answer

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

Find the values of c such that the area of the region bounded by the parabolas $y=x^{2}-c^{2} \text { and } y=c^{2}-x^{2}$ is $27$ .

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. $x^{2}+(y-1)^{2}=1^{2}$ about the y-axis

Find the work done in pushing a car a distance of 14 m while exerting a constant force of 370 N.

Find the area of the region bounded by the given curves. $y=\cos x, y=\sin 3 x, x=0, x=\frac{\pi}{2}$

Find the volume common to two spheres, each with radius r = $12$ if the center of each sphere lies on the surface of the other sphere.

The base of S is the parabolic region $\left\{(x, y) \mid x^{2} \leq y \leq 4\right\} .$ Cross-sections perpendicular to the y axis are squares. Find the volume of S.

Use the method of disks or washers, or 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. $y=2 \sqrt{x}$ , $y=x-3$ , $y=0$ , the x-axis

Find the volume of a cap of a sphere with radius r = $100$ and height h = 3 9 .

Use the method of disks or washers, or 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 = $x^{2}$ , y = 2x - 1, y = 4; the y-axis

Find the average value of the function $z(t)=3 \cos (t)$ on the interval $\left[0, \frac{\pi}{2}\right]$ .

Find the volume of a pyramid with height 4 and base an equilateral triangle with side a = 4 .

Find the area of the region bounded by the curves. $y=\cos x, y=\sin 3 x, x=0, x=\frac{\pi}{2}$

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. $y=3\left(x^{2}+4\right), y=3\left(12-x^{2}\right) ; \text { about } y=-3$

Find the average value of the function $f(t)=9 t \sin \left(t^{2}\right)$ on the interval $[0,20]$ . Round your answer to 3 decimal places.

The height of a monument is $20$ m. A horizontal cross-section at a distance x meters from the top is an equilateral triangle with side $\frac{x}{4}$ meters. Find the volume of the monument.

Find the volume of the solid obtained by rotating about the x-axis the region under the curve $y=\frac{1}{x}$ from x = 7 to x = 8 .

Find the area of the shaded region.

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 = $\frac{1}{x}$ , y = 0, x = 2, x = 5; 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

Find the values of c such that the area of the region bounded by the parabolas y=x2−c2 and y=c2−x2y=x^{2}-c^{2} \text { and } y=c^{2}-x^{2}y=x2−c2 and y=c2−x2 is 272727 .

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. x2+(y−1)2=12x^{2}+(y-1)^{2}=1^{2}x2+(y−1)2=12 about the y-axis

Find the work done in pushing a car a distance of 14 m while exerting a constant force of 370 N.

Find the area of the region bounded by the given curves. y=cos⁡x,y=sin⁡3x,x=0,x=π2y=\cos x, y=\sin 3 x, x=0, x=\frac{\pi}{2}y=cosx,y=sin3x,x=0,x=2π​

Find the volume common to two spheres, each with radius r = 121212 if the center of each sphere lies on the surface of the other sphere.

The base of S is the parabolic region {(x,y)∣x2≤y≤4}.\left\{(x, y) \mid x^{2} \leq y \leq 4\right\} .{(x,y)∣x2≤y≤4}. Cross-sections perpendicular to the y axis are squares. Find the volume of S.

Find the volume of a cap of a sphere with radius r = 100100100 and height h = 3 9 .

Use the method of disks or washers, or 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 = x2x^{2}x2 , y = 2x - 1, y = 4; the y-axis

Find the average value of the function z(t)=3cos⁡(t)z(t)=3 \cos (t)z(t)=3cos(t) on the interval [0,π2]\left[0, \frac{\pi}{2}\right][0,2π​] .

Find the volume of a pyramid with height 4 and base an equilateral triangle with side a = 4 .

Find the area of the region bounded by the curves. y=cos⁡x,y=sin⁡3x,x=0,x=π2y=\cos x, y=\sin 3 x, x=0, x=\frac{\pi}{2}y=cosx,y=sin3x,x=0,x=2π​

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y=3(x2+4),y=3(12−x2); about y=−3y=3\left(x^{2}+4\right), y=3\left(12-x^{2}\right) ; \text { about } y=-3y=3(x2+4),y=3(12−x2); about y=−3

Find the average value of the function f(t)=9tsin⁡(t2)f(t)=9 t \sin \left(t^{2}\right)f(t)=9tsin(t2) on the interval [0,20][0,20][0,20] . Round your answer to 3 decimal places.

The height of a monument is 202020 m. A horizontal cross-section at a distance x meters from the top is an equilateral triangle with side x4\frac{x}{4}4x​ meters. Find the volume of the monument.

Find the volume of the solid obtained by rotating about the x-axis the region under the curve y=1xy=\frac{1}{x}y=x1​ from x = 7 to x = 8 .

Find the area of the shaded region.

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 = 1x\frac{1}{x}x1​ , y = 0, x = 2, x = 5; 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

Find the values of c such that the area of the region bounded by the parabolas $y=x^{2}-c^{2} \text { and } y=c^{2}-x^{2}$ is $27$ .

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. $x^{2}+(y-1)^{2}=1^{2}$ about the y-axis

Find the area of the region bounded by the given curves. $y=\cos x, y=\sin 3 x, x=0, x=\frac{\pi}{2}$

Find the volume common to two spheres, each with radius r = $12$ if the center of each sphere lies on the surface of the other sphere.

The base of S is the parabolic region $\left\{(x, y) \mid x^{2} \leq y \leq 4\right\} .$ Cross-sections perpendicular to the y axis are squares. Find the volume of S.

Find the volume of a cap of a sphere with radius r = $100$ and height h = 3 9 .

Use the method of disks or washers, or 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 = $x^{2}$ , y = 2x - 1, y = 4; the y-axis

Find the average value of the function $z(t)=3 \cos (t)$ on the interval $\left[0, \frac{\pi}{2}\right]$ .

Find the area of the region bounded by the curves. $y=\cos x, y=\sin 3 x, x=0, x=\frac{\pi}{2}$

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. $y=3\left(x^{2}+4\right), y=3\left(12-x^{2}\right) ; \text { about } y=-3$

Find the average value of the function $f(t)=9 t \sin \left(t^{2}\right)$ on the interval $[0,20]$ . Round your answer to 3 decimal places.

The height of a monument is $20$ m. A horizontal cross-section at a distance x meters from the top is an equilateral triangle with side $\frac{x}{4}$ meters. Find the volume of the monument.

Find the volume of the solid obtained by rotating about the x-axis the region under the curve $y=\frac{1}{x}$ from x = 7 to x = 8 .

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 = $\frac{1}{x}$ , y = 0, x = 2, x = 5; the y-axis