Question 1

Which of the following best describes the main advantage of the cylindrical shell method over the washer/disk method when finding the volume of a solid of revolution?

Accepted Answer

It allows integration with respect to the original variable, avoiding the need to solve for the other variable.

Question 2

Given the region bounded by $$y = 2x^2$ - x^3$ and y = 0$, rotated about the y$-axis, what is the correct setup for the volume using the shell method?

Accepted Answer

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

Question 3

When rotating the region bounded by $$y = x$$ and $$y = x^2$ about the y$-axis, what is the height of a typical cylindrical shell at position x$?

Accepted Answer

$$x - x^2$

Question 4

A region under $$y = \sqrt{x}$$ from $$x = 0 to$$ $$x = 1 is $$rotated about the $$x-$$axis. What is the radius of a typical shell when using the shell method with respect to $$y$$?

Accepted Answer

$$y$$

Question 5

Which of the following integrals correctly represents the volume of the solid obtained by rotating the region bounded by $$y = x - x^2$ and y = 0$ about the line x = 2$?

Accepted Answer

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

Question 6

In the shell method formula $$V = \int_a^b 2\pi x f(x) dx$$, what does $$2\pi x$$ represent geometrically?

Accepted Answer

The circumference of the shell

Question 7

Suppose you are asked to find the volume of a solid formed by rotating the region between $$y = x^2$ and y = 0$ from x = 0$ to x = 1$ about the y$-axis. Which method is more efficient and why?

Accepted Answer

Shell method, because it avoids solving for $$x in $$terms of $$y.$$

Question 8

A cylindrical shell has radius $$x$$, height $$f(x)$$, and thickness $$dx. $$What is the formula for the volume of this shell?

Accepted Answer

$$2\pi x f(x) dx$$

Question 9

If the region bounded by $$y = x$$ and $$y = x^2$ is rotated about the y$-axis, what are the limits of integration for the shell method?

Accepted Answer

$$x = 0 to$$ $$x = 1$$

Question 10

Which of the following real-world scenarios could be modeled using the cylindrical shell method?

Accepted Answer

Calculating the volume of a hollow pipe formed by rotating a region around an axis.

Question 11

When rotating the region under $$y = \sqrt{x}$$ from $$x = 0 to$$ $$x = 1$$ about the $$x-$$axis, what is the correct integral for the volume using the shell method?

Accepted Answer

$$2\pi \$$int_0$$^1 y \cdot y^2$ dy$$

Question 12

Why is the shell method preferred when the washer method requires solving $$y = 2x^2$ - x^3$ for x$ in terms of y$?

Accepted Answer

Because the shell method allows integration with respect to $$x$$, avoiding the need to solve for $$x.$$

Question 13

If a region is rotated about the $$y-$$axis and described by $$y = f(x)$$, $$y = 0$$, $$x = a$$, $$x = b$$, what is the general shell method formula for the volume?

Accepted Answer

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

Question 14

A student claims that the shell method cannot be used when rotating about a vertical line other than the $$y-$$axis. Is this correct?

Accepted Answer

No, the shell method can be adapted for any vertical axis by adjusting the radius.

Question 15

What is the geometric interpretation of the thickness $$dx in $$the shell method?

Accepted Answer

It is the width of the shell.

Applications of Definite Integrals: The Method of Cylindrical Shells

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