Question 1

Which of the following best describes the disk method for finding the volume of a solid of revolution?

Accepted Answer

It approximates the volume by summing up the volumes of disks perpendicular to the axis of revolution.

Question 2

Given a region bounded above by $$y = \sqrt{\sin x}$$ and below by the x-axis for $$0 \leq x \leq \pi$$, what is the integral expression for the volume of the solid formed by revolving this region about the x-axis using the disk method?

Accepted Answer

$$V = \pi \int_{0}^{\pi} (\sqrt{\sin x})^2$ dx$$

Question 3

A solid is formed by revolving the region bounded by $$y = x^2$ and y$$ = \sqrt{x}$$ about the x-axis from x = 0$ to x = 1$. Which method and formula should be used to find its volume?

Accepted Answer

Washer method: $$V = \pi \int_{0}^{1} \left[(\sqrt{x})^2$ - (x^2$)^2$$\right] dx$

Question 4

When using the disk method to find the volume of a solid of revolution about the y-axis, which of the following is the correct general formula?

Accepted Answer

$$V = \pi \int_{c}^{d} [R(y)]^2 dy$$

Question 5

A cylindrical water tank has a radius of 5 meters and a height of 10 meters. Using the disk method, what is the volume of the tank?

Accepted Answer

$$V = \pi (5)^2$ (10) = 250\pi$$

Question 6

Which of the following real-world objects can be best modeled by a solid of revolution generated by revolving a rectangle about one of its sides?

Accepted Answer

A right circular cylinder

Question 7

Suppose a region is bounded by $$y = x^2$ + 1$$, $$y = 0$$, $$x = 0$$, and $$x = 1. If $$this region is revolved about the y-axis, which of the following best describes the setup for the volume using the washer method?

Accepted Answer

$$V = \pi \int_{0}^{1} [1^2$ - 0^2$] dx + \pi \int_{1}^{2} [1^2$ - (\sqrt{y-1})^2$] dy$$

Question 8

A solid has a base in the $$xy-$$plane bounded by $$y = 1 - \frac{x}{2}$$ and $$y = -1 + \frac{x}{2}$$ for $$0 \leq x \leq 2. If $$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} (2 - x)^2$

Question 9

Which of the following integrals gives the volume of a solid whose cross-sectional area perpendicular to the $$x-$$axis is $$A(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$V = \int_{a}^{b} A(x) dx$$

Question 10

A region bounded by $$y = x^2$ and y$$ = \sqrt{x}$$ is revolved about the x-axis. What is the physical meaning of the inner and outer radii in the washer method for this problem?

Accepted Answer

The inner radius is $$y = x^2$ and the outer radius is y$$ = \sqrt{x}$$, representing the distances from the x-axis to the curves.

Question 11

Which of the following is a correct application of the disk method for finding the volume of a solid generated by revolving the region under $$y = f(x)$$ from $$x = a to$$ $$x = b$$ about the x-axis?

Accepted Answer

$$V = \pi \int_{a}^{b} [f(x)]^2 dx$$

Question 12

A machine part is modeled as a solid of revolution generated by revolving the region under $$y = 2x$$ from $$x = 0 to$$ $$x = 3$$ about the x-axis. What is the volume of the part?

Accepted Answer

$$V = \pi \int_{0}^{3} (2x)^2$ dx = \pi \int_{0}^{3} 4x^2$ dx = 4\pi \left[\frac{x^3$}{3}\right]_{0}^{3} = 36\pi$$

Question 13

Which of the following best explains why the washer method is used instead of the disk method for some solids of revolution?

Accepted Answer

The region being revolved has a hole, so the volume must be calculated by subtracting the inner solid from the outer solid.

Question 14

A solid has a base in the $$xy-$$plane bounded by $$y = x^2$ and x = 1$, and cross sections perpendicular to the x-axis are squares. What is the integral for the volume of the solid?

Accepted Answer

$$V = \int_{0}^{1} (1 - x^2$)^2 dx$

Question 15

A region bounded by $$y = x^2$ + 1$$, $$y = 0$$, $$x = 0$$, and $$x = 1 is $$revolved about the y-axis. For $$1 \leq y \leq 2$$, what is the inner radius $$r(y) in $$terms of $$y$$?

Accepted Answer

$$r(y) = \sqrt{y - 1}$$

Question 16

Which of the following is the correct formula for the volume of a solid with known cross-sectional area $$A(y)$$ perpendicular to the y-axis from $$y = c to$$ $$y = d$$?

Accepted Answer

$$V = \int_{c}^{d} A(y) dy$$

Applications of Integration: Volumes of Solids Using Disk, Washer, and Cross Section Methods

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