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To find the volume of the solid obtained by rotating the region bounded by \( y = x^6 \) and \( y = 1 \) about the line \( y = 10 \), we use the method of washers. The washers' method involves integrating the area of circular washers, which differ in radius and are stacked along the axis of rotation.

### Steps to Calculate the Volume:
1. **Identify the bounds of integration:** From the equations \( y = x^6 \) and \( y = 1 \), we solve for \( x \) when \( y = 1 \) to find the bounds. Since \( x^6 = 1 \), \( x = 1 \) and \( x = -1 \) are the bounds of integration.

2. **Set up the integral for the washer method:** The volume \( V \) of the solid is given by:
   \[
   V = \pi \int_{a}^{b} [R_{\text{out}}^2(x) - R_{\text{in}}^2(x)] \, dx
   \]
   where \( R_{\text{out}}(x) \) and \( R_{\text{in}}(x) \) are the outer and inner radii of the washers, respectively.

   - The outer radius, \( R_{\text{out}}(x) \), is the distance from the line \( y = 10 \) to \( y = 1 \), which is \( 10 - 1 = 9 \).
   - The inner radius, \( R_{\text{in}}(x) \), is the distance from the line \( y = 10 \) to \( y = x^6 \), which is \( 10 - x^6 \).

3. **Write the integral:**
   \[
   V = \pi \int_{-1}^{1} [9^2 - (10 - x^6)^2] \, dx
   \]

Let's calculate this integral:

\[
V = \pi \int_{-1}^{1} [81 - (100 - 20x^6 + x^{12})] \, dx
\]

\[
V = \pi \int_{-1}^{1} [-19 + 20x^6 - x^{12}] \, dx
\]

Now, let's compute the integral to find the volume.

Find the volume of the solid obtained by rotating the region bounded by y = x^6, y = 1; about the line y = 10

Explore how to calculate the volume of a solid formed by rotating the region bounded by y = x^6 and y = 1 around the line y = 10. The solution utilizes the washer method in integral calculus, focusing on determining both the outer and inner radii for the integral setup. This problem is suitable for advanced mathematics students at the college or university level.

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