Use the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equation about the given line.

*y* = 6*x* − *x*2,      *y* = 0,

about the line *x* = 8

We are tasked with finding the volume of the solid generated by revolving the region bounded by the curves \( y = 6x - x^2 \) and \( y = 0 \) about the line \( x = 8 \). To achieve this, we will use the **shell method**.

### Step-by-Step Solution:

1. **Identify the region**:
   The curves that bound the region are:
   - \( y = 6x - x^2 \), a parabola opening downwards.
   - \( y = 0 \), the x-axis.

   We need to find the intersection points of these curves, i.e., solve:
   \[
   6x - x^2 = 0
   \]
   Factoring:
   \[
   x(6 - x) = 0
   \]
   So, \( x = 0 \) and \( x = 6 \).

2. **Set up the shell method formula**:
   The volume using the shell method is given by:
   \[
   V = 2\pi \int_{a}^{b} ( \text{radius} ) \times ( \text{height} ) \, dx
   \]
   where:
   - The **radius** is the horizontal distance from the curve to the line of rotation \( x = 8 \). This distance is \( 8 - x \).
   - The **height** of the shell is given by the value of \( y \), which is \( 6x - x^2 \).

   Therefore, the volume integral becomes:
   \[
   V = 2\pi \int_{0}^{6} (8 - x)(6x - x^2) \, dx
   \]

3. **Expand the integrand**:
   Let's expand the expression inside the integral:
   \[
   (8 - x)(6x - x^2) = 8(6x - x^2) - x(6x - x^2)
   \]
   Simplifying both terms:
   \[
   = 48x - 8x^2 - 6x^2 + x^3
   \]
   \[
   = 48x - 14x^2 + x^3
   \]

4. **Evaluate the integral**:
   Now, we integrate term by term:
   \[
   V = 2\pi \int_{0}^{6} (48x - 14x^2 + x^3) \, dx
   \]
   - The integral of \( 48x \) is \( 24x^2 \),
   - The integral of \( -14x^2 \) is \( -\frac{14}{3}x^3 \),
   - The integral of \( x^3 \) is \( \frac{x^4}{4} \).

   So, the volume is:
   \[
   V = 2\pi \left[ 24x^2 - \frac{14}{3}x^3 + \frac{x^4}{4} \right]_{0}^{6}
   \]

5. **Substitute the limits**:
   First, plug in \( x = 6 \):
   \[
   24(6)^2 - \frac{14}{3}(6)^3 + \frac{(6)^4}{4}
   = 24(36) - \frac{14}{3}(216) + \frac{1296}{4}
   = 864 - 1008 + 324 = 180
   \]
   When \( x = 0 \), all terms vanish.

   Therefore, the volume is:
   \[
   V = 2\pi \times 180 = 360\pi
   \]

### Final Answer:
The volume of the solid is \( \boxed{360\pi} \) cubic units.

Would you like further clarification on any part of the solution? Here are some related questions you can explore:

1. How does the shell method compare to the disk/washer method in this type of problem?
2. What if the line of rotation were \( x = -8 \)? How would the setup change?
3. How do you find the volume when the region is rotated about the y-axis using the shell method?
4. What if the parabola had a different equation, say \( y = 5x - x^2 \)?
5. How does the choice of the axis of rotation affect the resulting solid's shape?

**Tip**: When using the shell method, the radius is always the distance from the axis of rotation to the representative shell. Keep this in mind when setting up the problem!

Use the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equation y = 6x − x^2, y = 0, about the line x = 8.

Find the volume of a solid generated by revolving the region bounded by y = 6x - x^2 and y = 0 around the line x = 8 using the shell method. This detailed solution involves calculus concepts such as the shell method and solving quadratic functions, suitable for Grade 12 or college-level calculus.

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