Question 1

Which of the following integrals correctly represents the volume of a solid with cross-sectional area $$A(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$V = \int_a^b A(x)\,dx$$

Question 2

A right circular cylinder has radius $$r$$ and height $$h. $$What is its volume?

Accepted Answer

$$V = \pi r^2$ h$$

Question 3

The base of a solid is the region between $$y = 4\sin x$$ and the $$x-$$axis for $$0 \leq x \leq \pi. If $$cross-sections perpendicular to the $$x-$$axis are squares, what is the formula for the area of a typical cross-section at $$x$$?

Accepted Answer

$$A(x) = [4\sin x]^2$$

Question 4

A region bounded by $$y = x^2$ and y = 4$ is revolved about the x$-axis. Which method is most appropriate to find the volume?

Accepted Answer

Washer method

Question 5

Find the volume of the solid generated by revolving the region bounded by $$y = x^2$ and y = 4$ about the x$-axis.

Accepted Answer

$$V = \pi \$$int_0$$^2 [4 - x^2$]^2 dx$$

Question 6

A solid has a base in the $$xy-$$plane bounded by $$y = x$$ and $$y = x^2$ from x = 0$ to x = 1$. If cross-sections perpendicular to the x$-axis are equilateral triangles, what is the area of a typical cross-section at x$?

Accepted Answer

$$A(x) = \frac{\sqrt{3}}{4} [x - x^2$]^2$$

Question 7

Which of the following integrals gives the volume of a sphere of radius $$a$$ using the disk method?

Accepted Answer

$$V = \pi \int_{-a}^a (a^2$ - x^2$) dx$$

Question 8

A region bounded by $$y = x$$ and $$y = x^2$ is revolved about the y$-axis. Which method is most efficient for finding the volume?

Accepted Answer

Shell method

Question 9

A plane region is rotated about the $$x-$$axis to create a solid. The outer radius is $$R(x)$$ and the inner radius is $$r(x). $$What is the formula for the volume using the washer method?

Accepted Answer

$$V = \pi \int_a^b [R(x)^2$ - r(x)^2$] dx$$

Question 10

The region bounded by $$y = \sin x$$ and the $$x-$$axis from $$x = 0 to$$ $$x = \pi is $$revolved about the $$x-$$axis. What is the integral for the volume?

Accepted Answer

$$V = \pi \$$int_0$$^{\pi} [\sin x]^2 dx$$

Question 11

A wedge is cut from a right circular cylinder of radius $$3 by $$two planes, one perpendicular to the axis and the other crossing at $$45^\circ at $$the center. Which integral would you use to find the volume of the wedge?

Accepted Answer

$$V = \$$int_0$$^3 A(x) dx$$,\ where\ A(x)\ is\ the\ area\ of\ the\ cross-section\ at\ x$

Question 12

The region bounded by $$y = x^2$ and y = 1$ is revolved about the line y = 1$. What is the outer radius R(x$$)$$ for the washer method?

Accepted Answer

$$R(x) = 1$$

Question 13

Which of the following best describes the first step in calculating the volume of a solid by cross-sections?

Accepted Answer

Sketch the solid and a typical cross-section

Question 14

A region bounded by $$y = x$$ and $$y = x^2$ is revolved about the x$-axis. What is the correct integral for the volume?

Accepted Answer

$$V = \pi \$$int_0$$^1 [x^2$ - x^4$] dx$$

Question 15

The region bounded by $$y = x$$ and $$y = x^2$ is revolved about the y$-axis. What is the shell radius and height for the shell method?

Accepted Answer

Radius: $$x$$, Height: $$x - x^2$

Question 16

A solid has a base in the $$xy-$$plane bounded by $$y = x^2$ and y = 4$. If cross-sections perpendicular to the y$-axis are squares, what is the side length of a typical square at y$?

Accepted Answer

$$x = \sqrt{y}$$,\ so\ side\ length = \sqrt{y}$

Question 17

Which of the following is the correct formula for the volume of a right circular cone with base radius $$R$$ and height $$h$$?

Accepted Answer

$$V = \frac{1}{3}\pi R^2$ h$$

Question 18

A region bounded by $$y = \sin x$$ and the $$x-$$axis from $$x = 0 to$$ $$x = \pi is $$revolved about the $$y-$$axis. Which method is most efficient for finding the volume?

Accepted Answer

Shell method

Question 19

A region bounded by $$y = x^2$ and y = 4$ is revolved about the y$-axis. What is the correct integral for the volume using the shell method?

Accepted Answer

$$V = 2\pi \$$int_0$$^2 x (4 - x^2$) dx$$

Volumes of Solids Using Slicing, Disk, and Washer Methods

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