Find the volume of the solid obtained by rotating the region enclosed by x=0, y=1, x=y^7 about the line y=1 using the method of disks or washers

To find the volume of the solid obtained by rotating the region enclosed by the curves \(x = 0\), \(y = 1\), and \(x = y^7\) about the line \(y = 1\), we will use the washer method.

### Step-by-Step Solution:

1. **Understand the Problem:**
   - The region is bounded by \(x = 0\), \(y = 1\), and \(x = y^7\).
   - We are rotating this region around the line \(y = 1\).

2. **Set Up the Integral:**
   - The washer method formula for volume is:
   \[
   V = \int_{a}^{b} \pi \left( R^2 - r^2 \right) \, dx
   \]
   where \(R\) is the outer radius, and \(r\) is the inner radius of the washers.

3. **Determine the Radii:**
   - In this case, we rotate around \(y = 1\), so we need to express everything in terms of \(x\) and find the radii from the line \(y = 1\).
   - The outer radius \(R(x)\) is the distance from the line \(y = 1\) to the curve \(x = 0\) (which is a horizontal line at \(y = 0\)):
     \[
     R(x) = 1 - 0 = 1.
     \]
   - The inner radius \(r(x)\) is the distance from the line \(y = 1\) to the curve \(x = y^7\). Since we have \(x = y^7\), we can express \(y\) in terms of \(x\):
     \[
     y = x^{1/7}.
     \]
     Thus, the inner radius is:
     \[
     r(x) = 1 - x^{1/7}.
     \]

4. **Set Up the Limits of Integration:**
   - The region is enclosed from \(x = 0\) to \(x = 1\) (since \(y = 1\) implies \(x = 1^7 = 1\)).

5. **Formulate the Integral:**
   - The volume integral is:
   \[
   V = \int_{0}^{1} \pi \left[(1)^2 - (1 - x^{1/7})^2\right] \, dx.
   \]
   - Simplify the integrand:
   \[
   V = \int_{0}^{1} \pi \left[1 - (1 - 2x^{1/7} + x^{2/7})\right] \, dx.
   \]
   - This simplifies further to:
   \[
   V = \int_{0}^{1} \pi \left(2x^{1/7} - x^{2/7}\right) \, dx.
   \]

6. **Integrate:**
   - Now, integrate term by term:
   \[
   V = \pi \left( \int_{0}^{1} 2x^{1/7} \, dx - \int_{0}^{1} x^{2/7} \, dx \right).
   \]
   - Compute each integral:
     \[
     \int x^{1/7} \, dx = \frac{x^{8/7}}{8/7} = \frac{7}{8}x^{8/7},
     \]
     \[
     \int x^{2/7} \, dx = \frac{x^{9/7}}{9/7} = \frac{7}{9}x^{9/7}.
     \]
   - Applying the limits from 0 to 1:
     \[
     V = \pi \left( 2 \cdot \frac{7}{8} \cdot 1^{8/7} - \frac{7}{9} \cdot 1^{9/7} \right).
     \]
   - Simplify the expression:
     \[
     V = \pi \left( \frac{14}{8} - \frac{7}{9} \right) = \pi \left( \frac{63}{36} - \frac{28}{36} \right).
     \]
   - This simplifies to:
     \[
     V = \pi \cdot \frac{35}{36} = \frac{35\pi}{36}.
     \]

### Final Answer:
\[
\boxed{\frac{35\pi}{36}}
\]

Would you like more details or have any questions?

### Related Questions:
1. How would the volume change if the region were rotated about the line \(y = -1\)?
2. What is the volume if we rotate the same region around the y-axis instead of the line \(y = 1\)?
3. Can the shell method be used to solve this problem? If so, how?
4. What is the area of the region enclosed by \(x = 0\), \(y = 1\), and \(x = y^7\)?
5. How do you find the centroid of the region enclosed by \(x = 0\), \(y = 1\), and \(x = y^7\)?

**Tip:** Always make sure to correctly identify the outer and inner radii when setting up integrals for the washer method in volume problems.

Find the volume of the solid obtained by rotating the region enclosed by x=0, y=1, x=y^7 about the line y=1 using the method of disks or washers.

Learn how to find the volume of the solid obtained by rotating the region enclosed by x=0, y=1, and x=y^7 about the line y=1 using the washer method. This detailed solution covers key calculus concepts, including the washer method and integration. Suitable for undergraduate calculus students.

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