Questions: Find the volume of the solid generated by revolving the region bounded by y=4 sqrt(x) and the lines y=4 and x=0 about a. the x-axis. b. the y-axis. c. the line y=4. d. the line x=1. a. The volume of the solid generated by revolving the region bounded by y=4 sqrt(x) and the lines y=4 and x=0 about the x-axis is 8 pi cubic units. b. The volume of the solid generated by revolving the region bounded by y=4 sqrt(x) and the lines y=4 and x=0 about the y-axis is 4 pi/5 cubic units. c. The volume of the solid generated by revolving the region bounded by y=4 sqrt(x) and the lines y=4 and x=0 about the line y=4 is 8 pi/3 cubic units. d. The volume of the solid generated by revolving the region bounded by y=4 sqrt(x) and the lines y=4 and x=0 about the line x=1 is 32 pi/15 cubic units.

Find the volume of the solid generated by revolving the region bounded by y=4 sqrt(x) and the lines y=4 and x=0 about a. the x-axis. b. the y-axis. c. the line y=4. d. the line x=1. a. The volume of the solid generated by revolving the region bounded by y=4 sqrt(x) and the lines y=4 and x=0 about the x-axis is 8 pi cubic units. b. The volume of the solid generated by revolving the region bounded by y=4 sqrt(x) and the lines y=4 and x=0 about the y-axis is 4 pi/5 cubic units. c. The volume of the solid generated by revolving the region bounded by y=4 sqrt(x) and the lines y=4 and x=0 about the line y=4 is 8 pi/3 cubic units. d. The volume of the solid generated by revolving the region bounded by y=4 sqrt(x) and the lines y=4 and x=0 about the line x=1 is 32 pi/15 cubic units.
Transcript text: Find the volume of the solid generated by revolving the region bounded by $y=4 \sqrt{x}$ and the lines $y=4$ and $x=0$ about a. the $x$-axis. b. the $y$-axis. c. the line $y=4$. d. the line $x=1$. a. The volume of the solid generated by revolving the region bounded by $y=4 \sqrt{x}$ and the lines $y=4$ and $x=0$ about the $x$-axis is $8 \pi$ cubic units. b. The volume of the solid generated by revolving the region bounded by $y=4 \sqrt{x}$ and the lines $y=4$ and $x=0$ about the $y$-axis is $\frac{4 \pi}{5}$ cubic units. c. The volume of the solid generated by revolving the region bounded by $y=4 \sqrt{x}$ and the lines $y=4$ and $x=0$ about the line $y=4$ is $\frac{8 \pi}{3}$ cubic units. d. The volume of the solid generated by revolving the region bounded by $y=4 \sqrt{x}$ and the lines $y=4$ and $x=0$ about the line $x=1$ is $\frac{32 \pi}{15}$ cubic units.
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Solution

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Solution Steps

Step 1: Volume Around the x-axis

To find the volume of the solid generated by revolving the region bounded by \( y = 4\sqrt{x} \) and the lines \( y = 4 \) and \( x = 0 \) about the \( x \)-axis, we use the disk method. The volume is given by:

\[ V = \int_{0}^{1} \pi (4\sqrt{x})^2 \, dx = \int_{0}^{1} 16\pi x \, dx \]

Calculating this integral, we find:

\[ V = 8\pi \]

Step 2: Volume Around the y-axis

For the volume generated by revolving the same region about the \( y \)-axis, we apply the shell method. The volume is given by:

\[ V = \int_{0}^{1} 2\pi x (4\sqrt{x}) \, dx = \int_{0}^{1} 8\pi x^{3/2} \, dx \]

Evaluating this integral yields:

\[ V = \frac{16\pi}{5} \]

Step 3: Volume Around the Line \( y = 4 \)

To find the volume of the solid generated by revolving the region about the line \( y = 4 \), we use the washer method. The volume is given by:

\[ V = \int_{0}^{1} \pi \left(4^2 - (4 - 4\sqrt{x})^2\right) \, dx \]

Calculating this integral results in:

\[ V = \frac{40\pi}{3} \]

Final Answer

The volumes for each part are as follows:

  • a. Volume around the \( x \)-axis: \( 8\pi \)
  • b. Volume around the \( y \)-axis: \( \frac{16\pi}{5} \)
  • c. Volume around the line \( y = 4 \): \( \frac{40\pi}{3} \)

Thus, the final answers are:

\[ \boxed{8\pi}, \quad \boxed{\frac{16\pi}{5}}, \quad \boxed{\frac{40\pi}{3}} \]

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