Question 1

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

Accepted Answer

It uses cylindrical shells formed by revolving a rectangle parallel to the axis of revolution.

Question 2

Given the region bounded by $$y = x - x^3$ and the x-axis for 0$$ \leq x \leq 1$$, what is the correct shell method integral to find the volume when revolved about the y-axis?

Accepted Answer

$$2\pi \$$int_0$$^1 x(x - x^3$) dx$$

Question 3

What is the formula for the volume of a cylindrical shell with average radius $$p$$, height $$h$$, and thickness $$w$$?

Accepted Answer

$$2\pi p h w$$

Question 4

When is the shell method generally preferable to the disk/washer method?

Accepted Answer

When the representative rectangle is parallel to the axis of revolution.

Question 5

A region bounded by $$y = x^2$ + 1$$, $$y = 0$$, $$x = 0$$, and $$x = 1 is $$revolved about the y-axis. What is the shell method integral for its volume?

Accepted Answer

$$2\pi \$$int_0$$^1 x(x^2$ + 1) dx$$

Question 6

Which of the following is the general shell method formula for a vertical axis of revolution?

Accepted Answer

$$V = 2\pi \int_a^b p(x) h(x) dx$$

Question 7

A water tank is shaped like a solid of revolution formed by revolving the region bounded by $$y = 2 - x^2$ and the x-axis about the y-axis. What is the shell method integral for the volume?

Accepted Answer

$$2\pi \$$int_0$$^{\sqrt{2}} x(2 - x^2$) dx$$

Question 8

Which real-world object could best be modeled using the shell method for volume calculation?

Accepted Answer

A cylindrical water tank with a hollow core

Question 9

If a region is revolved about a horizontal axis, which variable is typically used for integration in the shell method?

Accepted Answer

$$y$$

Question 10

A solid is formed by revolving the region bounded by $$y = x$$ and $$y = x^2$ about the y-axis. What is the shell method integral for the volume?

Accepted Answer

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

Question 11

Which of the following is a key difference between the disk/washer method and the shell method?

Accepted Answer

The disk/washer method uses rectangles perpendicular to the axis of revolution, while the shell method uses rectangles parallel to the axis.

Question 12

A machine part is shaped like a hollow cylinder with inner radius $$r_1$$, outer radius $$r_2$$, and height $$h. $$What is its volume?

Accepted Answer

$$\pi h (r_2$^2 - r_1$^2)$$

Question 13

Which of the following integrals represents the volume of a solid generated by revolving the region between $$y = x$$ and $$y = 0$$ from $$x = 0 to$$ $$x = 2$$ about the y-axis using the shell method?

Accepted Answer

$$2\pi \$$int_0$$^2 x(x) dx$$

Question 14

A region bounded by $$y = \sqrt{x}$$, $$y = 0$$, and $$x = 4 is $$revolved about the y-axis. What is the shell method integral for the volume?

Accepted Answer

$$2\pi \$$int_0$$^4 x\sqrt{x} dx$$

Question 15

Which of the following best explains why the shell method can be more efficient than the disk/washer method in some cases?

Accepted Answer

It can reduce the number of required integrals when the region is bounded by functions of $$x$$ and revolved about the y-axis.

Applications of Integration: The Shell Method for Volumes of Solids of Revolution

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