Show all your work to receive full credit. Write your answers as complete sentences.

Use the disk method to find the volume of the solid of revolution generated by rotating the
region between the graph of and the -axis over the interval about the -
axis.

Ask by Hanson Ortiz. in the United States

Jan 19,2025

Question

Show all your work to receive full credit. Write your answers as complete sentences.

Use the disk method to find the volume of the solid of revolution generated by rotating the
region between the graph of and the -axis over the interval about the -
axis.

Ask by Hanson Ortiz. in the United States

Jan 19,2025

UpStudy AI · Accepted Answer

To find the volume of the solid of revolution generated by rotating the region between the graph of $$ f(x) = \sqrt{x} $$ and the $$ x $$-axis over the interval $$ [0,8] $$ about the $$ x $$-axis using the disk method, we will follow these steps:

1. **Understand the Disk Method:**
   
   The disk method involves integrating the area of circular disks perpendicular to the axis of rotation. The volume $$ V $$ is given by:
   $$
   V = \pi \int_{a}^{b} [f(x)]^2 \, dx
   $$
   where $$ f(x) $$ is the function representing the radius of the disk at position $$ x $$, and $$ [a, b] $$ is the interval of integration.

2. **Set Up the Integral:**
   
   In this problem, $$ f(x) = \sqrt{x} $$, and we're rotating around the $$ x $$-axis from $$ x = 0 $$ to $$ x = 8 $$. Therefore, the radius of each disk is $$ \sqrt{x} $$, and the area of each disk is $$ \pi (\sqrt{x})^2 = \pi x $$.

The volume integral becomes:
   $$
   V = \pi \int_{0}^{8} x \, dx
   $$

3. **Evaluate the Integral:**
   
   Compute the integral of $$ x $$ with respect to $$ x $$:
   $$
   \int_{0}^{8} x \, dx = \left. \frac{x^2}{2} \right|_{0}^{8} = \frac{8^2}{2} - \frac{0^2}{2} = \frac{64}{2} - 0 = 32
   $$

4. **Multiply by $$ \pi $$:**
   
   Multiply the result of the integral by $$ \pi $$ to obtain the volume:
   $$
   V = \pi \times 32 = 32\pi
   $$

5. **Final Answer:**
   
   Therefore, the volume of the solid of revolution is $$ 32\pi $$ cubic units.
The volume of the solid of revolution is $$ 32\pi $$ cubic units.

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Use the disk method to find the volume of the solid of revolution generated by rotating the
region between the graph of and the -axis over the interval about the -
axis.

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Show all your work to receive full credit. Write your answers as complete sentences. Use the disk method to find the volume of the solid of revolution generated by rotating the region between the graph of and the -axis over the interval about the - axis.

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Show all your work to receive full credit. Write your answers as complete sentences.

Use the disk method to find the volume of the solid of revolution generated by rotating the
region between the graph of and the -axis over the interval about the -
axis.