evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the y-axis.
y = x^(3/2)
y = 8
x = 0

To find the volume of the solid generated by revolving the region bounded by \( y = x^{3/2} \), \( y = 8 \), and \( x = 0 \) about the **y-axis**, we will use the **method of cylindrical shells**. 

Here’s a step-by-step solution:

### Step 1: Express \( x \) in terms of \( y \)
We are given the function \( y = x^{3/2} \), so we first solve for \( x \) in terms of \( y \).

\[
y = x^{3/2} \implies x = y^{2/3}
\]

### Step 2: Set up the formula for the volume using the method of cylindrical shells
The formula for the volume using the method of cylindrical shells is:

\[
V = 2\pi \int_{y_{\text{min}}}^{y_{\text{max}}} \left(\text{radius}\right) \times \left(\text{height}\right) \, dy
\]

- The **radius** is the distance from the y-axis to the function, which is \( x = y^{2/3} \).
- The **height** is the difference in \( x \)-coordinates of the boundaries, which is \( y^{2/3} - 0 \).
- \( y_{\text{min}} \) is the lower bound of \( y \), which is 0 (from \( x = 0 \)), and \( y_{\text{max}} \) is 8.

Thus, the volume integral becomes:

\[
V = 2\pi \int_{0}^{8} y^{2/3} \cdot 1 \, dy
\]

### Step 3: Evaluate the integral
Now, evaluate the definite integral:

\[
V = 2\pi \int_{0}^{8} y^{2/3} \, dy
\]

Using the power rule for integration:

\[
\int y^{n} \, dy = \frac{y^{n+1}}{n+1} + C
\]

So,

\[
V = 2\pi \left[ \frac{y^{5/3}}{5/3} \right]_{0}^{8}
\]

Simplify:

\[
V = 2\pi \left( \frac{3}{5} \right) \left[ y^{5/3} \right]_{0}^{8}
\]

Substitute the limits of integration:

\[
V = 2\pi \left( \frac{3}{5} \right) \left( 8^{5/3} - 0^{5/3} \right)
\]

Since \( 8^{5/3} = (2^3)^{5/3} = 2^5 = 32 \), we have:

\[
V = 2\pi \left( \frac{3}{5} \right) \times 32
\]

\[
V = 2\pi \times \frac{96}{5}
\]

\[
V = \frac{192\pi}{5}
\]

### Final Answer:
The volume of the solid is:

\[
V = \frac{192\pi}{5} \ \text{cubic units}.
\]

Would you like more details or have any questions?

Here are five related questions:
1. How would the method change if we revolved the region around the x-axis instead?
2. What is the geometric interpretation of the cylindrical shells method?
3. How would the integral change if the curve was \( y = x^{1/2} \) instead?
4. What happens to the volume if the region is revolved around a vertical line other than the y-axis?
5. How does the volume change if the upper bound of \( y \) changes from 8 to another value?

**Tip**: Always ensure that the limits of integration match the axis of revolution and variable you are working with when setting up your integral.

Evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the y-axis. y = x^(3/2), y = 8, x = 0

Explore how to calculate the volume of a solid generated by revolving the region bounded by y = x^(3/2), y = 8, and x = 0 about the y-axis using the cylindrical shells method. This step-by-step solution involves definite integrals and calculus concepts suitable for college-level students.

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