Use cylindrical shells to find the volume of the solid obtained by rotating the region pounded by y = x² , y = 0, and x = 1, about the y-axis.

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## Calculating Volume Using Cylindrical Shells

To find the volume of a solid obtained by rotating the specified region around the y-axis, we will use the method of cylindrical shells.

### Problem Statement
Use cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the curves \( y = x^2 \), \( y = 0 \), and \( x = 1 \) about the y-axis.

### Setup
Given the boundaries:
- \( y = x^2 \) (a parabola)
- \( y = 0 \) (the x-axis)
- \( x = 1 \) (a vertical line)

### Solution Steps
1. **Determine the height and radius of a typical shell:**
   - Height (h) of the shell at any point \( x \) is given by the function: \( h = y = x^2 \).
   - Radius (r) of the shell: Since we are rotating around the y-axis, the radius is equal to the x-coordinate \( x \).

2. **Express the volume element:**
   - The volume element (dV) of a cylindrical shell is: \( dV = 2\pi x \cdot (x^2) \cdot dx = 2\pi x^3 \, dx \).

3. **Integrate to find the total volume:**
   - The limits of integration are from \( x = 0 \) to \( x = 1 \).
   - The volume \( V = \int_{0}^{1} 2\pi x^3 \, dx \).

4. **Calculate the integral:**
   - \( V = 2\pi \int_{0}^{1} x^3 \, dx \).
   - \( V = 2\pi \left[ \frac{x^4}{4} \right]_{0}^{1} \).
   - \( V = 2\pi \left( \frac{1}{4} - 0 \right) \).
   - \( V = \frac{\pi}{2} \).

### Final Answer
\[ V = \frac{\pi}{2} \]
Transcribed Image Text:## Calculating Volume Using Cylindrical Shells To find the volume of a solid obtained by rotating the specified region around the y-axis, we will use the method of cylindrical shells. ### Problem Statement Use cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the curves \( y = x^2 \), \( y = 0 \), and \( x = 1 \) about the y-axis. ### Setup Given the boundaries: - \( y = x^2 \) (a parabola) - \( y = 0 \) (the x-axis) - \( x = 1 \) (a vertical line) ### Solution Steps 1. **Determine the height and radius of a typical shell:** - Height (h) of the shell at any point \( x \) is given by the function: \( h = y = x^2 \). - Radius (r) of the shell: Since we are rotating around the y-axis, the radius is equal to the x-coordinate \( x \). 2. **Express the volume element:** - The volume element (dV) of a cylindrical shell is: \( dV = 2\pi x \cdot (x^2) \cdot dx = 2\pi x^3 \, dx \). 3. **Integrate to find the total volume:** - The limits of integration are from \( x = 0 \) to \( x = 1 \). - The volume \( V = \int_{0}^{1} 2\pi x^3 \, dx \). 4. **Calculate the integral:** - \( V = 2\pi \int_{0}^{1} x^3 \, dx \). - \( V = 2\pi \left[ \frac{x^4}{4} \right]_{0}^{1} \). - \( V = 2\pi \left( \frac{1}{4} - 0 \right) \). - \( V = \frac{\pi}{2} \). ### Final Answer \[ V = \frac{\pi}{2} \]
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