Find the volume of the solid formed by rotating the region enclosed by x=8y and x=y^3 (with y>=0) about the y-axis.

To find the volume of the solid formed by rotating the region enclosed by \( x = 8y \) and \( x = y^3 \) (with \( y \geq 0 \)) about the y-axis, we can use the method of cylindrical shells.

### Step 1: Identify the Region of Interest
The curves are:
- \( x = 8y \)
- \( x = y^3 \)

The region is enclosed between these curves for \( y \geq 0 \).

### Step 2: Find the Intersection Points
To determine the limits of integration, find the points where the curves intersect by setting \( x = 8y \) equal to \( x = y^3 \):
\[
8y = y^3
\]
\[
y^3 - 8y = 0
\]
\[
y(y^2 - 8) = 0
\]
This gives us \( y = 0 \) or \( y^2 = 8 \). Therefore, the intersection points are \( y = 0 \) and \( y = 2\sqrt{2} \).

### Step 3: Set Up the Integral Using the Shell Method
The volume of the solid of revolution using the shell method is given by:
\[
V = 2\pi \int_{a}^{b} \text{(radius)} \times \text{(height)} \, dy
\]
Here:
- The radius of each cylindrical shell is \( y \) (the distance from the y-axis).
- The height of each shell is the difference between the x-values of the curves \( x = 8y \) and \( x = y^3 \), so the height is \( 8y - y^3 \).

The limits of integration are from \( y = 0 \) to \( y = 2\sqrt{2} \).

Thus, the integral becomes:
\[
V = 2\pi \int_{0}^{2\sqrt{2}} y(8y - y^3) \, dy
\]

### Step 4: Simplify the Integral
Distribute \( y \) inside the integral:
\[
V = 2\pi \int_{0}^{2\sqrt{2}} (8y^2 - y^4) \, dy
\]

### Step 5: Evaluate the Integral
Integrate term by term:
\[
V = 2\pi \left[ \frac{8y^3}{3} - \frac{y^5}{5} \right]_{0}^{2\sqrt{2}}
\]
Substitute the limits of integration:
\[
V = 2\pi \left( \frac{8(2\sqrt{2})^3}{3} - \frac{(2\sqrt{2})^5}{5} \right)
\]

### Step 6: Simplify the Expression
Calculate each term:
\[
(2\sqrt{2})^3 = 8\sqrt{2}, \quad (2\sqrt{2})^5 = 32 \times 8\sqrt{2} = 256\sqrt{2}
\]
Thus, we have:
\[
V = 2\pi \left( \frac{8 \times 8\sqrt{2}}{3} - \frac{256\sqrt{2}}{5} \right)
\]
\[
V = 2\pi \left( \frac{64\sqrt{2}}{3} - \frac{256\sqrt{2}}{5} \right)
\]

### Step 7: Find a Common Denominator and Combine the Terms
To combine the fractions, use a common denominator of 15:
\[
V = 2\pi \left( \frac{320\sqrt{2}}{15} - \frac{768\sqrt{2}}{15} \right)
\]
\[
V = 2\pi \times \frac{-448\sqrt{2}}{15}
\]

### Final Answer
The volume of the solid is:
\[
\boxed{-\frac{896\pi\sqrt{2}}{15}}
\]
Since volume cannot be negative, we take the absolute value:
\[
\boxed{\frac{896\pi\sqrt{2}}{15}}
\]

Would you like any details or have further questions?

Here are 5 related questions to expand your understanding:
1. How would the setup change if we rotated around the x-axis instead of the y-axis?
2. Can you derive the same volume using the disk method?
3. How does the volume compare if the curves were rotated about \( y = 2 \)?
4. What is the significance of the negative sign that appeared in the calculation?
5. What happens if the curves \( x = 8y \) and \( x = y^3 \) are shifted horizontally before rotation?

**Tip:** When using the shell method, carefully consider the orientation of the radius and height of each cylindrical shell relative to the axis of rotation.

Learn how to find the volume of a solid of revolution formed by rotating the region enclosed by x=8y and x=y^3 about the y-axis. This problem utilizes integral calculus, specifically the cylindrical shells method. Suitable for college-level calculus students.

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