Question 1

Which of the following best describes the method for finding the volume of a solid using cross-sections perpendicular to the x-axis?

Accepted Answer

Integrate the area of the cross-section over the interval.

Question 2

A pyramid is 3 meters high with a square base of side 3 meters. What is the volume of the pyramid?

Accepted Answer

Question 3

Given a solid with base bounded by the x-axis and $$y = 6x - x^2$, and cross-sections perpendicular to the x-axis are equilateral triangles, what is the general formula for the area of a cross-section at x$?

Accepted Answer

$$A(x) = \frac{\sqrt{3}}{4} [6x - x^2$]^2$$

Question 4

Cavalieri’s Principle states that two solids with equal altitudes and identical cross-sectional areas at each height have the same:

Accepted Answer

Volume

Question 5

What is the formula for the volume of a solid of revolution generated by revolving $$y = f(x)$$ about the x-axis using the disk method?

Accepted Answer

$$V = \int_a^b \pi [f(x)]^2 dx$$

Question 6

A region above the x-axis and below $$y = \sqrt{a^2$ - x^2$} is $$revolved about the x-axis. What is the resulting solid and its volume?

Accepted Answer

A sphere of radius $$a$$ with volume $$V = \frac{4}{3}\pi a^3$

Question 7

When using the washer method to find the volume of a solid of revolution, what is the formula for the area of a cross-section at $$x$$?

Accepted Answer

$$A(x) = \pi [R(x)]^2 - \pi [r(x)]^2$$

Question 8

A region bounded by $$y = 4 - 3x^2$ and y$$ = $$x^2 is $$revolved about the x-axis. What is the integral expression for the volume?

Accepted Answer

$$V = \int_{-1}^{1} \pi [(4 - 3x^2$)^2$$ - $$(x^2)^2$] dx$$

Question 9

Which of the following is a correct application of the washer method for revolving a region about the y-axis?

Accepted Answer

$$V = \int_c^d \pi ([R(y)]^2 - [r(y)]^2) dy$$

Question 10

A solid has a base bounded by the x-axis, y-axis, and $$2x + 3y = 6. $$Cross-sections perpendicular to the x-axis are semicircles. What is the diameter of a cross-section at $$x$$?

Accepted Answer

$$d = \frac{6 - 2x}{3}$$

Question 11

If a region is revolved about the line $$y = 1$$, what is the radius of a disk at $$x$$ for the region between $$y = f(x)$$ and $$y = 1$$?

Accepted Answer

$$|f(x) - 1|$$

Question 12

Which of the following is NOT a requirement for using the disk method?

Accepted Answer

The region must be bounded by two different curves.

Question 13

A right tetrahedron has lengths 5, 4, and 3 in the x-, y-, and z-directions, respectively. What is the area of a cross-section parallel to the xy-plane at height $$z$$?

Accepted Answer

$$A(z) = \frac{20}{9}(3-z)^2$

Question 14

Which principle allows us to conclude that two solids with identical cross-sectional areas at every height and equal altitudes have the same volume?

Accepted Answer

Cavalieri’s Principle

Question 15

A region bounded by $$y = 3$$, the y-axis, $$y = 1$$, and $$y = 4 is $$revolved about the y-axis. What is the integral for the volume?

Accepted Answer

$$V = \$$int_1$$^4 \pi (3^2$) dy$$

Question 16

When revolving a region about $$y = 2$$, what is the outer radius $$R(x)$$ for a region bounded above by $$y = 2$$ and below by $$y = f(x)$$?

Accepted Answer

$$R(x) = 2 - f(x)$$

Question 17

Which of the following best describes the difference between the disk and washer methods?

Accepted Answer

The disk method is used when the region is bounded by the axis of revolution; the washer method is used when there is a gap between the region and the axis.

Applications of Definite Integrals: Volumes Using Cross-Sections

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