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

Study Guide - Practice Questions

Study Guide - Flashcards