Question 1

A right circular cone has height $$5$$ and base radius $$2x. $$What is the area of the cross-section cut by the plane $$x = c$$ perpendicular to the $$x-$$axis?

Accepted Answer

$$4\pi c^2$

Question 2

Given a solid whose base is the region bounded by $$y = 1 - x^2$ and the x$-axis, and whose cross-sections perpendicular to the x$-axis are squares, what is the volume of the solid?

Accepted Answer

$$\dfrac{16}{15}$$

Question 3

For the same base region as above ($$y = 1 - x^2$ and the x$-axis$$), if the cross-sections perpendicular to the $$x$-axis are semicircles with diameter in the base, what is the volume of the solid?

Accepted Answer

Question 4

Let $$R be $$the region bounded by $$y = \sqrt{x}$$, $$x = 4$$, and the $$x-$$axis. What is the volume of the solid formed when $$R is $$revolved about the $$x-$$axis?

Accepted Answer

$$8\pi$$

Question 5

Which of the following integrals correctly represents the volume of a solid of revolution using the disc method, where the cross-sectional radius is $$R(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

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

Question 6

A region bounded by $$y = x^2$ and y = 4$ is revolved about the x$-axis. Using the washer method, what are the outer and inner radii for a typical washer at position x$?

Accepted Answer

Outer radius: $$4$$, Inner radius: $$x^2$$

Question 7

Given the region bounded by $$y = x$$ and $$y = x^2$ from x = 0$ to x = 1$, revolved about the x$-axis, which integral gives the volume of the resulting solid?

Accepted Answer

$$\$$int_0$$^1 \pi (x^2$ - x^4$) dx$$

Question 8

A solid has cross-sections perpendicular to the $$x-$$axis that are equilateral triangles, and the base is the region bounded by $$y = x$$ and $$y = 2x$$ from $$x = 0 to$$ $$x = 1. $$What is the area of a cross-section at $$x$$?

Accepted Answer

$$\dfrac{\sqrt{3}}{4} (2x - x)^2$

Question 9

Suppose you want to estimate the volume of a cone of height $$5$$ and base radius $$2x by $$slicing it into $$n$$ discs of equal thickness. What is the thickness of each slice?

Accepted Answer

$$\dfrac{5}{n}$$

Question 10

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

Accepted Answer

Subtracting the volume of the inner solid from the outer solid using $$V = \int_a^b \pi (R(x)^2$ - r(x)^2$) dx$$

Question 11

A region bounded by $$y = x^2$ and y = 4$ is revolved about the x$-axis. What is the volume of the resulting solid between x = 0$ and x = 2$?

Accepted Answer

$$\dfrac{128\pi}{5}$$

Question 12

If a solid has cross-sections perpendicular to the $$x-$$axis that are circles with radius $$r(x)$$, what is the formula for the volume of the solid from $$x = a to$$ $$x = b$$?

Accepted Answer

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

Question 13

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

Accepted Answer

$$(1 - x^2$)^2$

Question 14

Which of the following is the correct setup for the volume of a solid with base bounded by $$y = x^2$ and y = 1$, and cross-sections perpendicular to the x$-axis that are semicircles?

Accepted Answer

Question 15

A region bounded by $$y = x$$ and $$y = x^2$ is revolved about the y$-axis. Which method would you use to find the volume and why?

Accepted Answer

Washer method, because the region forms washers with varying inner and outer radii as $$y$$ increases.

Question 16

If a solid has cross-sections perpendicular to the $$x-$$axis that are rectangles of height $$h(x)$$ and width $$w(x)$$, what is the formula for the volume from $$x = a to$$ $$x = b$$?

Accepted Answer

$$V = \int_a^b h(x) w(x) dx$$

Question 17

A solid has a base in the $$xy-$$plane bounded by $$y = x^2$ and y = 4$, and cross-sections perpendicular to the x$-axis are equilateral triangles. What is the area of a cross-section at x$?

Accepted Answer

$$\dfrac{\sqrt{3}}{4} (4 - x^2$)^2$

Question 18

Which of the following is the correct formula for the volume of a solid of revolution using the washer method, where the outer radius is $$R(x)$$ and the inner radius is $$r(x)$$?

Accepted Answer

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

Volumes of Solids with Known Cross-Sections: Disc and Washer Methods

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