Question 1

Find the volume of the described solid.&#10;

Accepted Answer

The area of each cross section is $$A(x) = (2 \sqrt { x } + 2 \sqrt { x } )^2 = 16x$$So, the volume of the solid is$$V = \int_0^9 16x \ dx = \left[ 8x^2 \right]_0^9 = 648$$

Question 2

Find the volume of the solid generated by revolving the shaded region about the given axis.&#10;

Accepted Answer

To use the disk method, we need to consider a vertical axis of rotation. We can see that the region is bounded to the left and right by the y-axis and the line x = –4, respectively. So, if we revolve this region about the x-axis, we will generate a series of disks of thickness ∆x and radius y, where y will vary with x as the function $y = 16 - x^2$. Thus, the volume of each disk can be expressed as $\pi y^2 \, \mathrm{d}x$. Integrating this expression from –4 to 4 will give us the volume of the solid:
$$\int_{-4}^4 \pi (16 - x^2)^2 \, \mathrm{d}x = \frac{7168}{15} \pi$$

Question 3

Find the volume of the described solid.&#10;

Accepted Answer

D

Question 4

Find the volume of the solid generated by revolving the shaded region about the given axis.&#10;

Accepted Answer

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

Question 5

Find the volume of the solid generated by revolving the shaded region about the given axis.&#10;

Accepted Answer

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Question 6

Find the volume of the solid generated by revolving the shaded region about the given axis.&#10;

Accepted Answer

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Question 7

Find the volume of the solid generated by revolving the shaded region about the given axis.&#10;

Accepted Answer

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Question 8

Find the volume of the solid generated by revolving the shaded region about the given axis.&#10;

Accepted Answer

The region to be revolved around the $x$-axis is bounded by the curves $y=16-x^2$ and the $x$-axis. We are revolving this region around the $x$-axis, so the disk method will be appropriate.

The equation $y=16-x^2$ is a parabola opening downwards with vertex at the origin. To find the limits of integration, we set the expression $16-x^2$ equal to 0 to find that the endpoints of integration are at $x=4$ and $x=-4$.

The radius of the disk for a given $x$-value is simply $y$, so $r=16-x^2$.

Thus, the volume of the solid is given by:

$$V = \int_{-4}^4 \pi (16-x^2)^2 \, dx = \frac{8192}{15}\pi$$

Therefore, the answer is (D).

Question 9

Find the volume of the solid generated by revolving the shaded region about the given axis.&#10;

Accepted Answer

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

Question 10

Find the volume of the solid generated by revolving the shaded region about the given axis.&#10;

Accepted Answer

To use the method of cylindrical shells, we need to express the equation in terms of $x$. Squaring both sides of $y=3\sqrt{\sin x}$ gives us $$y^2=9\sin x.$$Re-arranging, we get $\sin x=\frac{y^2}{9}.$ When we revolve the shaded region about the $x$-axis, we generate cylindrical shells of radius $y=3\sqrt{\sin x}$ and height $dx$. Therefore, the volume of a typical cylindrical shell is given by $$dV=2\pi y\cdot dx=2\pi(3\sqrt{\sin x})\cdot dx.$$The volume of the solid is given by$$\int_{0}^{\pi}2\pi(3\sqrt{\sin x})\cdot dx=6\pi\int_{0}^{\pi}\sqrt{\sin x}\cdot dx.$$To integrate $\sqrt{\sin x}$, we make the substitution $u=\cos x,$ so $du=-\sin x\,dx$. When $x=0,$ we have $u=1.$ When $x=\pi,$ we have $u=-1.$ Therefore, we get $$\int_{0}^{\pi}\sqrt{\sin x}\cdot dx=\int_{1}^{-1}\sqrt{1-u^2}\cdot (-du)=\int_{-1}^{1}\sqrt{1-u^2}\cdot du=\frac{\pi}{2}.$$The last integral evaluates to $\frac{\pi}{2}$ because it is the area of a quarter of a unit circle. Substituting this back into our expression for the volume, we get $$6\pi\cdot\frac{\pi}{2}=3\pi^2.$$Finally, the volume generated is $3\pi^2$, so the correct answer is $\boxed{	extbf{(A)}}$.

Question 11

Find the volume of the described solid.&#10;

Accepted Answer

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Question 12

Find the volume of the described solid.&#10;

Accepted Answer

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Question 13

Find the volume of the described solid.&#10;

Accepted Answer

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Question 14

Find the volume of the solid generated by revolving the shaded region about the given axis.&#10;

Accepted Answer

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Question 15

Find the volume of the solid generated by revolving the shaded region about the given axis.&#10;

Accepted Answer

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Question 16

Find the volume of the described solid.&#10;

Accepted Answer

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Question 17

Find the volume of the described solid.&#10;

Accepted Answer

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Question 18

Find the volume of the described solid.&#10; &#10;A) 90&#10;B) 18&#10;C) 72&#10;D) 144

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Question 19

Find the volume of the described solid.&#10;

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Question 20

Find the volume of the described solid.&#10;

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Question 21

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.&#10;

Accepted Answer

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Question 22

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.&#10;

Accepted Answer

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Question 23

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.&#10;

Accepted Answer

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Question 24

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.&#10;

Accepted Answer

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Question 25

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.&#10;

Accepted Answer

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Question 26

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.&#10;

Accepted Answer

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Question 27

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.&#10;

Accepted Answer

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Question 28

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.&#10;y = - 7x + 14, y = 7x, x = 0

Accepted Answer

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Question 29

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.&#10;

Accepted Answer

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Question 30

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.&#10;

Accepted Answer

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Question 31

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.&#10;

Accepted Answer

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Question 32

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.&#10;y = x, y = 0, x = 2, x = 4

Accepted Answer

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Question 33

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.&#10;y = x + 2, y = 0, x = -2, x = 6

Accepted Answer

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Question 34

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.&#10;

Accepted Answer

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Question 35

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.&#10;

Accepted Answer

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Question 36

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.&#10;y = 3x, y = 3, x = 0

Accepted Answer

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Question 37

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.&#10;

Accepted Answer

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Question 38

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.&#10;

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Question 39

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.&#10;

Accepted Answer

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Question 40

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.&#10;

Accepted Answer

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Question 41

Find the volume of the solid generated by revolving the region about the y-axis.&#10;

Accepted Answer

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Question 42

Find the volume of the solid generated by revolving the region about the y-axis.&#10;

Accepted Answer

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Question 43

Find the volume of the solid generated by revolving the region about the given line.&#10;

Accepted Answer

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Question 44

Find the volume of the solid generated by revolving the region about the y-axis.&#10;The region enclosed by the triangle with vertices (0, 0), (4, 0), (4, 2)

Accepted Answer

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Question 45

Find the volume of the solid generated by revolving the region about the y-axis.&#10;

Accepted Answer

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Question 46

Find the volume of the solid generated by revolving the region about the y-axis.&#10;

Accepted Answer

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Question 47

Find the volume of the solid generated by revolving the region about the given line.&#10;

Accepted Answer

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Question 48

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.&#10;

Accepted Answer

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Question 49

Find the volume of the solid generated by revolving the region about the y-axis.&#10;

Accepted Answer

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Question 50

Find the volume of the solid generated by revolving the region about the given line.&#10;

Accepted Answer

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Question 51

Find the volume of the solid generated by revolving the region about the given line.&#10;

Accepted Answer

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Question 52

Find the volume of the solid generated by revolving the region about the given line.&#10;

Accepted Answer

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Question 53

Find the volume of the solid generated by revolving the region about the given line.&#10;

Accepted Answer

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Question 54

Find the volume of the solid generated by revolving the region about the y-axis.&#10;

Accepted Answer

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Question 55

Find the volume of the solid generated by revolving the region about the given line.&#10;

Accepted Answer

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Question 56

Find the volume of the solid generated by revolving the region about the given line.&#10;

Accepted Answer

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Question 57

Find the volume of the solid generated by revolving the region about the y-axis.&#10;

Accepted Answer

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Question 58

Find the volume of the solid generated by revolving the region about the y-axis.&#10;The region enclosed by the triangle with vertices (1, 0), (1, 2), (3, 2)

Accepted Answer

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Question 59

Find the volume of the solid generated by revolving the region about the given line.&#10;

Accepted Answer

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Question 60

Find the volume of the solid generated by revolving the region about the given line.&#10;

Accepted Answer

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Question 61

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis.&#10;

Accepted Answer

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Question 62

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis.&#10;

Accepted Answer

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Question 63

Find the volume of the solid generated by revolving the region about the y-axis.&#10;

Accepted Answer

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Question 64

Solve the problem.&#10;A frustum of a right circular cone has a height of 10 m, a base of radius 5m, and a top of radius 4m. Find its volume.

Accepted Answer

The answer of Solve the problem.&#10;A frustum of a right...

Deck 7: Applications of Definite Integrals