Let A be the region bounded by y=x^3, y=8, and x=0. Set up the integral for the volume of each of the following solids. 1) Revolving A about the y-axis

To set up the integral for the volume of the solid formed by revolving the region \( A \) around the \( y \)-axis, we first need to understand the region and how it changes when rotated.

### Region Description
The region \( A \) is bounded by:
- \( y = x^3 \) (a cubic function),
- \( y = 8 \) (a horizontal line),
- \( x = 0 \) (the \( y \)-axis).

This region is located in the first quadrant, from \( x = 0 \) to \( x = 2 \) because at \( x = 2 \), \( y = 2^3 = 8 \).

### Setting up the Integral

Since the solid is formed by revolving around the \( y \)-axis, we will use the **shell method**. In the shell method, the volume \( V \) is given by:

\[
V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx,
\]

where \( x \) is the radius of the shell and \( f(x) \) is the height of the shell.

In this case:
- \( x \) is the radius of a shell at a given \( x \).
- The height of a shell is the distance from \( y = x^3 \) to \( y = 8 \), which is \( 8 - x^3 \).

The limits of integration are from \( x = 0 \) to \( x = 2 \).

Therefore, the volume integral is:

\[
V = 2\pi \int_{0}^{2} x (8 - x^3) \, dx.
\]

### Simplifying the Integral

Distribute \( x \) inside the integral:

\[
V = 2\pi \int_{0}^{2} (8x - x^4) \, dx.
\]

### The Integral Expression

So, the integral set up for the volume of the solid when the region \( A \) is revolved about the \( y \)-axis is:

\[
V = 2\pi \int_{0}^{2} (8x - x^4) \, dx.
\]

Would you like me to compute this integral, provide more details, or do you have any questions?

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**Related Questions:**

1. How would the integral change if the region were revolved around the \( x \)-axis instead?
2. What is the volume of the solid formed when revolving the region around the line \( x = 3 \)?
3. How does the shell method differ from the disk/washer method for this problem?
4. Can you set up the integral using the washer method for revolution around the \( y \)-axis?
5. What would be the volume if the region were revolved around the line \( y = -1 \)?

**Tip:** When setting up integrals for volumes of revolution, visualize the region and the axis of rotation to determine the appropriate method (shell or washer).

Let A be the region bounded by y=x^3, y=8, and x=0. Set up the integral for the volume of each of the following solids: 1) Revolving A about the y-axis.

Set up the integral to find the volume of the solid formed by revolving the region bounded by y=x^3, y=8, and x=0 about the y-axis using the shell method. This problem involves key calculus concepts and is suitable for students in Grades 11-12.

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