Question 1

Given the region $$R$$ bounded by $$y = x^2$ + 1$$, $$y = 1$$, and $$x = 2$$, which of the following correctly expresses the radius and height of a typical shell when using the shell method to compute the volume of the solid obtained by revolving $$R$$ about the vertical line $$x = 3$$?

Accepted Answer

Radius: $$3 - x$$, Height: $$x^2$$

Question 2

For the same region $$R as $$above, what is the correct shell method integral (do not evaluate) for the volume when revolving $$R$$ about $$x = 3$$?

Accepted Answer

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

Question 3

When revolving the region $$R$$ bounded by $$y = x^2$ + 1$$, $$y = 1$$, and $$x = 2$$ about the horizontal line $$y = -1$$ using the shell method, what is the correct expression for the radius of a typical shell?

Accepted Answer

$$y + 1$$

Question 4

Which of the following integrals correctly represents the shell method setup for the volume of the solid obtained by revolving $$R$$ about $$y = -1$$?

Accepted Answer

$$\$$int_1$$^5 2\pi (y + 1)(2 - \sqrt{y - 1}) \, dy$$

Question 5

Consider the triangular region $$R$$ bounded by $$y = x$$, $$y = 12 - 2x$$, and $$y = \frac{3}{2} - \frac{x}{2}. $$When revolving $$R$$ about $$x = 0$$ using the shell method, what is the correct expression for the height of a shell for $$1 \leq x \leq 4$$?

Accepted Answer

$$\frac{3x}{2} - \frac{3}{2}$$

Question 6

For the same region $$R$$ and axis $$x = 0$$, which of the following is the correct shell method integral for the volume for $$1 \leq x \leq 4$$?

Accepted Answer

$$\$$int_1$$^4 2\pi x \left( \frac{3x}{2} - \frac{3}{2} \right) dx$$

Question 7

When revolving the triangular region $$R$$ about $$x = 8$$, what is the radius of a typical shell for $$4 \leq x \leq 7$$?

Accepted Answer

$$8 - x$$

Question 8

For the same region $$R$$ and axis $$x = 8$$, what is the correct shell method integral for the volume for $$4 \leq x \leq 7$$?

Accepted Answer

$$\$$int_4$$^7 2\pi (8 - x) \left( \frac{21}{2} - \frac{3x}{2} \right) dx$$

Question 9

For the region $$R$$ and revolving about $$y = -3$$, what is the correct expression for the height of a shell for $$-2 \leq y \leq 1$$?

Accepted Answer

$$3 + 3y/2$$

Question 10

Which of the following integrals correctly represents the shell method setup for the volume of the solid obtained by revolving $$R$$ about $$y = -3$$ for $$1 \leq y \leq 4$$?

Accepted Answer

$$\$$int_1$$^4 2\pi (y + 3) (6 - 3y/2) dy$$

Question 11

When revolving the region $$R$$ about $$y = 5$$, what is the radius of a typical shell for $$-2 \leq y \leq 1$$?

Accepted Answer

$$5 - y$$

Question 12

For the same region $$R$$ and axis $$y = 5$$, what is the correct shell method integral for the volume for $$-2 \leq y \leq 1$$?

Accepted Answer

$$\int_{-2}^1 2\pi (5 - y) (3 + 3y/2) dy$$

Question 13

Suppose you are asked to find the volume of a solid generated by revolving a region bounded by $$y = f(x)$$, $$y = g(x)$$, $$x = a$$, and $$x = b$$ about a vertical line $$x = c$$ using the shell method. What is the general form of the shell method integral?

Accepted Answer

$$\int_a^b 2\pi |c - x| [f(x) - g(x)] dx$$

Question 14

A medical imaging device creates a 3D model by revolving the region under $$y = x^2$ from x = 0$ to x = 2$ about the y$-axis. Which shell method integral gives the volume of the resulting solid?

Accepted Answer

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

Question 15

A chemical engineer needs to calculate the volume of a tank formed by revolving the region between $$y = x$$ and $$y = x^2$ from x = 0$ to x = 1$ about the y$-axis. Which shell method integral is correct?

Accepted Answer

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

Question 16

A region is bounded by $$y = x^2$ + 1$$, $$y = 1$$, and $$x = 2. If $$this region is revolved about the $$y-$$axis, which method would be most efficient to compute the volume?

Accepted Answer

Shell method, integrating with respect to $$x$$

Shell Method for Volumes of Solids of Revolution

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