Question 1

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

Accepted Answer

The volume of the solid can be found using the method of disks, where the volume $V$ is given by the integral $$V = \int_{a}^{b} A(x) \, dx,$$ where $A(x)$ is the area of a cross-section of the solid perpendicular to the $x$-axis. In this case, the cross-sections are circular disks with diameters from $y = \cot(x)$ to $y = \csc(x)$. The radius $r$ of such a disk is half the difference between $\csc(x)$ and $\cot(x)$, so $r = \frac{1}{2}(\csc(x) - \cot(x))$. The area of a disk is $A(x) = \pi r^2$, so we have $$A(x) = \pi \left(\frac{1}{2}(\csc(x) - \cot(x))\right)^2.$$ The volume is then $$V = \int_{\pi/6}^{\pi/2} \pi \left(\frac{1}{2}(\csc(x) - \cot(x))\right)^2 \, dx.$$ Solving this integral gives the volume of the solid, which matches the expression in option D.

Question 2

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

Accepted Answer

D

Question 3

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

Accepted Answer

The area of a cross section at $x$ is $$A(x) = \pi \left[ \left( \frac { 50 - x ^ { 2 } } { 2 } \right) ^ { 2 } - \left( \frac { x ^ { 2 } } { 2 } \right) ^ { 2 } \right] = \pi \left( 50 x ^ { 2 } - 25 x ^ { 4 } \right).$$So the volume of the solid is$$\begin{split} V & = \int _ { - 5 } ^ { 5 } \pi \left( 50 x ^ { 2 } - 25 x ^ { 4 } \right) d x \\ & = \pi \left[ \frac { 50 x ^ { 3 } } { 3 } - 5 x ^ { 5 } \right] _ { - 5 } ^ { 5 } \\ & = \frac { 10000 } { 3 } \pi \end{split}$$

Question 4

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

Accepted Answer

The cross-sectional area of the solid at a given value of $x$ is a square with side length equal to the distance between the two parabolas at that $x$. This distance is $2\sqrt{x} - (-2\sqrt{x}) = 4\sqrt{x}$. Therefore, the area of a cross-section is $(4\sqrt{x})^2 = 16x$. To find the volume of the solid, we integrate this area from $x = 0$ to $x = 7$:$$\text{Volume} = \int_{0}^{7} 16x \, dx = \left[8x^2\right]_{0}^{7} = 8(7)^2 = 8 \times 49 = 392.$$

Question 5

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

Accepted Answer

To revolve the shaded region about the $x$-axis, we need to integrate with respect to $y$. First, we need to solve for $x$ in terms of $y$:

$y = 4 - x^2$

$x^2 = 4-y$

$x = \pm \sqrt{4-y}$

The shaded region is bounded by the $y$-axis and the parabola $y=4-x^2$, so we can set up the integral as follows:

$$V = \int_{0}^{4} \pi x^2 dy$$

Since we know that $x = \sqrt{4-y}$, we can substitute:

$$V = \int_{0}^{4} \pi (\sqrt{4-y})^2 dy$$

$$V = \int_{0}^{4} \pi (4-y) dy$$

$$V = \pi \left[4y - \frac{y^2}{2}\right]_{0}^{4}$$

$$V = \pi \left[(16-8) - (0-0)\right]$$

$$V = \frac{8}{15} \pi \cdot 16 = \frac{256}{15}\pi$$

Therefore, the answer is D, $\frac{256}{15}\pi$.

Question 6

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

Accepted Answer

The radius of the cross-sectional circle is $r = \frac { 7 } { \sqrt { 4 + x ^ { 2 } } }$. So, the area of the cross section is $$A(x) = \pi r ^ { 2 } = \pi \left( \frac { 7 } { \sqrt { 4 + x ^ { 2 } } } \right) ^ { 2 } = \frac { 49 \pi } { 4 + x ^ { 2 } }$$Thus, the volume of the solid is$$V = \int _ { - 2 } ^ { 2 } A(x) \ dx = \int _ { - 2 } ^ { 2 } \frac { 49 \pi } { 4 + x ^ { 2 } } \ dx = \left[ 49 \pi \arctan \frac { x } { 2 } \right] _ { - 2 } ^ { 2 } = 49 \pi \left( \frac { \pi } { 4 } \right) = \frac { 49 } { 4 } \pi ^ { 2 }$$

Question 7

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 8

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

Accepted Answer

We know that the region between the graph of $y=\sqrt{3x}$ and the $x$-axis from $x=0$ to $x=3$ creates a triangular shape. When revolved around the $y$-axis, this shape will create a cone and we can find the volume of this cone using the formula $\frac{1}{3}\pi r^2h$. The radius ($r$) is the distance from the $y$-axis to the right side of the triangle, which is $3$. The height ($h$) is the distance from the $x$-axis to the top of the triangle, which is $\sqrt{3(3)}=3\sqrt{3}$. 
$$\frac{1}{3}\pi (3)^2(3\sqrt{3})=\frac{27}{5}\pi$$
Therefore, the volume of the solid generated by revolving the shaded region about the $y$-axis is $\frac{27}{5}\pi$, which is answer choice $\boxed{\textbf{(D)}}$.

Question 9

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

Accepted Answer

The volume of the solid can be calculated using the formula for the volume of a solid of revolution, specifically using the method of disks or washers. Here, the cross-sections are semicircles with diameters from $y = -\sqrt{9 - x^2}$ to $y = \sqrt{9 - x^2}$, which means the radius of each semicircle is $\frac{\sqrt{9 - x^2}}{2}$. The area of each semicircle is $\frac{1}{2} \pi r^2 = \frac{1}{2} \pi \left(\frac{\sqrt{9 - x^2}}{2}\right)^2$. Integrating this area from $x = -3$ to $x = 3$ gives the volume of the solid. The integral simplifies to $\int_{-3}^{3} \frac{1}{2} \pi \left(\frac{9 - x^2}{4}\right) dx$, which evaluates to $18 \pi$.

Question 10

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

Accepted Answer

The volume of the solid can be calculated using the method of cross-sections. The area of an isosceles right triangle with leg length $l$ is $\frac{1}{2}l^2$. For a given value of $y$, the length of the leg of the triangle is the diameter of the slice of the disk at that $y$, which is $2\sqrt{4-y^2}$ because the equation of the disk is $x^2 + y^2 = 4$. Thus, the area of a cross-section is $\frac{1}{2}(2\sqrt{4-y^2})^2 = 2(4-y^2)$. The volume is the integral of the cross-sectional area from $y = -2$ to $y = 2$, which is $\int_{-2}^{2} 2(4-y^2) dy = \left[ 8y - \frac{2}{3}y^3 \right]_{-2}^{2} = \left(16 - \frac{16}{3}\right) - \left(-16 + \frac{16}{3}\right) = \frac{64}{3}$.

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;

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Question 13

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

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Question 15

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

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Question 16

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

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Question 17

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

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Question 18

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

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Question 19

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

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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;

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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;

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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;

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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;

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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;

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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;

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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;

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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;

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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;

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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;

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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;

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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;

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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;

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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;

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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;

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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;

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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;

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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;

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Question 41

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 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 y-axis.&#10;

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Question 44

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

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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 given line.&#10;

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Question 47

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

Accepted Answer

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Question 48

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

Accepted Answer

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Question 49

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

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Question 51

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

Accepted Answer

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Question 52

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

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

Solve the problem.&#10;

Accepted Answer

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Question 62

Solve the problem.&#10;

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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

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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Deck 6: Applications of Definite Integrals