Use the shell method to find the volume of the solid generated by revolving the region bounded by the line y=x+12 and the parabola y=x^2 about the line x=4

To find the volume of the solid generated by revolving the region bounded by the line \( y = x + 12 \) and the parabola \( y = x^2 \) about the line \( x = 4 \), we can use the shell method.

### Steps to Solve:

1. **Set up the functions**:  
   The region is bounded by \( y = x^2 \) (parabola) and \( y = x + 12 \) (line).

2. **Find the points of intersection**:  
   To find the points where the curves intersect, set the equations equal to each other:
   \[
   x^2 = x + 12
   \]
   Rearrange this equation:
   \[
   x^2 - x - 12 = 0
   \]
   Factor the quadratic:
   \[
   (x - 4)(x + 3) = 0
   \]
   So, the solutions are \( x = 4 \) and \( x = -3 \). These are the points of intersection.

3. **Set up the shell method integral**:  
   The shell method formula for volume when revolving around a vertical line (in this case, \( x = 4 \)) is:
   \[
   V = 2\pi \int_{a}^{b} (r_{\text{shell}})(h_{\text{shell}}) \, dx
   \]
   Where:
   - \( r_{\text{shell}} \) is the distance from the shell to the axis of rotation: \( r_{\text{shell}} = 4 - x \).
   - \( h_{\text{shell}} \) is the height of the shell: \( h_{\text{shell}} = (x + 12) - x^2 \).

4. **Set up the integral**:  
   Integrate from \( x = -3 \) to \( x = 4 \):
   \[
   V = 2\pi \int_{-3}^{4} (4 - x)\left[(x + 12) - x^2\right] \, dx
   \]
   Simplify the integrand:
   \[
   V = 2\pi \int_{-3}^{4} (4 - x)\left(-x^2 + x + 

Learn how to calculate the volume of a solid generated by revolving the region bounded by y=x+12 and y=x^2 about the line x=4 using the shell method. This problem involves key calculus concepts such as integration and volume of solids. Suitable for students in Grades 11-12.

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