https://file.aiquickdraw.com/mathbot/file/1729749772625of0921js.jpg

We are tasked with using the shell method to find the volume generated by revolving the region bounded by the curve \( x = 20y - y^2 \) and \( x = 0 \) about the x-axis.

### Steps for solving using the Shell Method:

The **shell method** involves integrating over the height of cylindrical shells formed by revolving the region around the x-axis. The formula for the volume of revolution using the shell method about the x-axis is:

\[
V = 2\pi \int_{y_{\text{min}}}^{y_{\text{max}}} (y \cdot \text{shell height}) \, dy
\]

1. **Determine the shell height**:
   - The height of each cylindrical shell is the horizontal distance between the two curves, which is the difference between \( x = 20y - y^2 \) and \( x = 0 \):
   \[
   \text{Shell height} = (20y - y^2) - 0 = 20y - y^2
   \]

2. **Set the limits of integration**:
   - To find the limits of integration for \( y \), set \( x = 20y - y^2 = 0 \), which gives:
   \[
   y(20 - y) = 0
   \]
   Thus, \( y = 0 \) and \( y = 20 \). So, the limits of integration are from \( y = 0 \) to \( y = 20 \).

3. **Set up the integral**:
   The volume integral is:
   \[
   V = 2\pi \int_{0}^{20} y(20y - y^2) \, dy
   \]

4. **Expand and integrate**:
   First, expand the integrand:
   \[
   y(20y - y^2) = 20y^2 - y^3
   \]
   Now, integrate term by term:
   \[
   V = 2\pi \int_{0}^{20} (20y^2 - y^3) \, dy
   \]
   The integral of \( 20y^2 \) is \( \frac{20y^3}{3} \) and the integral of \( y^3 \) is \( \frac{y^4}{4} \), so we have:
   \[
   V = 2\pi \left[ \frac{20y^3}{3} - \frac{y^4}{4} \right]_{0}^{20}
   \]

5. **Evaluate the integral**:
   Plugging in the limits of integration:
   \[
   V = 2\pi \left( \left[ \frac{20(20)^3}{3} - \frac{(20)^4}{4} \right] - \left[ \frac{20(0)^3}{3} - \frac{(0)^4}{4} \right] \right)
   \]
   Simplifying:
   \[
   V = 2\pi \left( \frac{20 \times 8000}{3} - \frac{160000}{4} \right)
   \]
   \[
   V = 2\pi \left( \frac{160000}{3} - 40000 \right)
   \]
   Find a common denominator:
   \[
   V = 2\pi \left( \frac{160000 - 120000}{3} \right) = 2\pi \left( \frac{40000}{3} \right)
   \]
   \[
   V = \frac{80000\pi}{3}
   \]

Thus, the volume of the solid generated by revolving the region is:

\[
V = \frac{80000\pi}{3} \, \text{cubic units}
\]

Would you like more details or have any other questions?

Here are five related questions you might explore:
1. How does the shell method differ from the disk method in calculating volumes?
2. Can the region be revolved around the y-axis using the shell method? How would the integral change?
3. What are the geometric interpretations of the terms in the volume formula for cylindrical shells?
4. How would the limits of integration change if the problem involved different boundaries?
5. Can we use the shell method to find the volume of a solid with a different curve, such as \( x = y^2 \)?

**Tip:** Always pay attention to the axis of revolution in shell or disk method problems—this determines the correct formula and variable of integration.

Use the shell method to find the volume generated by revolving the region bounded by x = 20y - y^2 and x = 0 about the x-axis.

Learn how to use the shell method to find the volume of a solid generated by revolving the region bounded by x = 20y - y^2 about the x-axis. This calculus problem involves integration techniques and applications of the shell method.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation