Question 1

Suppose the region bounded by $$f(x) = -(x-3)^2$ + 1$$ and the $$x-$$axis is revolved about the $$x-$$axis. Which of the following definite integrals correctly represents the volume of the resulting solid of revolution?

Accepted Answer

$$\int_{1}^{5} \pi \left( -(x-3)^2$ + 1 \$$right)^2$$ dx$$

Question 2

Evaluate the definite integral $$\int_{1}^{5} \pi \left( -(x-3)^2$ + 1 \$$right)^2$$ dx to $$find the volume of the solid generated by revolving the region bounded by $$f(x) = -(x-3)^2$ + 1$$ and the $$x-$$axis about the $$x-$$axis.

Accepted Answer

$$\frac{256\pi}{15}$$

Question 3

Suppose the region in Quadrant I bounded by $$f(x) = e^x$$, the $$y-$$axis, and the line $$y = 5 is $$revolved about the $$y-$$axis. Which of the following integrals represents the volume of the resulting solid of revolution?

Accepted Answer

$$\int_{0}^{5} \pi (\ln y)^2$ dy$$

Question 4

Suppose the region bounded by $$f(x) = e^x$$, the $$y-$$axis, and the line $$y = 5 is $$revolved about the line $$y = 5. $$Which method and integral setup is correct?

Accepted Answer

$$\int_{0}^{5} \pi (5 - y)^2$ dy$$

Question 5

Suppose the region bounded by $$f(y) = \frac{3}{y}$$ and the lines $$x = 2$$ and $$y = 6 is $$revolved about the line $$x = 2. $$Which of the following integrals represents the volume of the resulting solid of revolution?

Accepted Answer

$$\int_{2}^{6} \pi (2 - \frac{3}{y})^2$ dy$$

Question 6

Evaluate the definite integral $$\int_{2}^{6} \pi (2 - \frac{3}{y})^2$ dy to $$find the volume of the solid generated by revolving the region bounded by $$f(y) = \frac{3}{y}$$, $$x = 2$$, and $$y = 6$$ about $$x = 2.$$

Accepted Answer

$$\pi \left[ 4y - 12 \ln y + \frac{9}{y} \right]_{2}^{6}$$

Question 7

Suppose the region bounded by $$f(x) = \sqrt{x}$$ and $$f(x) = x^4$ is revolved about the line y = 4$. Which of the following integrals represents the volume of the resulting solid of revolution?

Accepted Answer

$$\int_{0}^{2} \pi \left[ (4 - \sqrt{x})^2$ - (4 - x^4$)^2$$ \right] dx$

Question 8

Suppose the region bounded by $$f(x) = \sqrt{x}$$ and $$f(x) = x^4$ is revolved about the line x = -1$. Which of the following integrals represents the volume of the resulting solid of revolution?

Accepted Answer

$$\int_{0}^{1} 2\pi (x + 1)(\sqrt{x} - x^4$) dx$$

Question 9

Which method is most appropriate for finding the volume of a solid generated by revolving a region bounded by two curves about a line that is not an axis?

Accepted Answer

Washer method

Question 10

A medical device is modeled by revolving the region between $$y = e^x$$ and $$y = 5$$ from $$x = 0 to$$ $$x = \ln 5$$ about the $$y-$$axis. Which integral gives the volume of the device?

Accepted Answer

$$\int_{0}^{\ln 5} \pi (5^2$ - (e^x)^2$) dx$$

Question 11

A region bounded by $$y = \sqrt{x}$$, $$y = x^4$, and x = 0$ is revolved about the y$-axis. Which method and setup is correct?

Accepted Answer

$$\int_{0}^{1} 2\pi x (\sqrt{x} - x^4$) dx$$

Question 12

Which of the following best describes the geometric meaning of the washer method in finding volumes of revolution?

Accepted Answer

It subtracts the volume of a smaller solid from a larger solid, both generated by revolution.

Question 13

A region bounded by $$y = \frac{3}{x}$$, $$x = 2$$, and $$y = 6 is $$revolved about $$x = 2. $$What is the outer radius in the washer method setup?

Accepted Answer

$$2$$

Question 14

Which of the following is a common error when setting up the washer method for a region revolved about a non-axis line?

Accepted Answer

Forgetting to adjust the radius to account for the distance from the axis of revolution.

Calculus Practice: Volumes of Solids of Revolution (Cross Sections, Washer, Disk Methods)

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