Find the volume of the solid obtained by rotating the region enclosed by
𝑥=0,𝑦=1,𝑥=𝑦3
about the line 𝑦=1
 using the method of disks or washers.
just show me the volume integral






To find the volume of the solid obtained by rotating the region around the line \( y = 1 \) using the washer method, we need to set up the volume integral. The given region is enclosed by \( x = 0 \), \( y = 1 \), and \( x = y^3 \).

We rotate around \( y = 1 \), and thus the washer method requires us to calculate the difference between the outer and inner radii of the washers. The outer radius is the distance from the axis of rotation to the function, and the inner radius is zero (because one side of the region touches the axis of rotation).

The volume integral is:

\[
V = \pi \int_{y=0}^{1} \left[ (1 - 0)^2 - (1 - y)^2 \right] \cdot x \, dy
\]

Since \( x = y^3 \), we substitute that in for \( x \):

\[
V = \pi \int_{y=0}^{1} \left[ (1^2) - (1 - y)^2 \right] \cdot y^3 \, dy
\]

This is the volume integral for the solid.

Do you need further clarifications or any details on how to solve this integral?

### Related questions:
1. How do you apply the washer method to more complex shapes?
2. What changes in the integral when rotating around a different axis?
3. How would you find the volume if the rotation axis was a vertical line?
4. Can you explain why the limits of integration are from 0 to 1?
5. How would the setup change if we used cylindrical shells instead?

### Tip:
When setting up the washer method, always check that the radius of the washer corresponds to the perpendicular distance from the curve to the axis of rotation.

Find the volume of the solid obtained by rotating the region enclosed by x=0, y=1, x=y^3 about the line y=1 using the method of disks or washers. Just show me the volume integral.

Discover how to find the volume of a solid formed by rotating the region enclosed by x=0, y=1, and x=y^3 around the line y=1. This problem uses the washer method and integral calculus, perfect for college-level calculus students.

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