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The problem asks to find the volume of the solid formed by revolving the region under the curve \( y = x^2 \) about the y-axis. This can be done using the method of cylindrical shells.

### Cylindrical Shells Method:

For a curve \( y = f(x) \) revolved around the y-axis, the formula for the volume \( V \) using cylindrical shells is:

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

Here:
- \( f(x) = x^2 \),
- The limits of integration are from \( x = 0 \) to \( x = 2 \) (based on the graph provided),
- We are revolving about the y-axis.

### Step-by-step solution:

1. **Set up the integral:**

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

2. **Evaluate the integral:**

First, integrate \( x^3 \):

\[
\int x^3 \, dx = \frac{x^4}{4}
\]

So, the volume integral becomes:

\[
V = 2\pi \left[ \frac{x^4}{4} \right]_0^2
\]

3. **Apply the limits:**

\[
V = 2\pi \left( \frac{2^4}{4} - \frac{0^4}{4} \right)
\]

\[
V = 2\pi \left( \frac{16}{4} \right) = 2\pi \times 4 = 8\pi
\]

### Final Answer:

The volume of the solid formed is \( 8\pi \) cubic units.

Would you like further details or have any questions?

Here are 5 related questions:
1. How does the volume formula differ when revolving around the x-axis?
2. Can we solve this problem using the disk method?
3. What happens if the curve is revolved around a different line, like \( x = 3 \)?
4. How does the cylindrical shell method apply to other curves?
5. Can this be generalized for any polynomial function?

**Tip**: The method of cylindrical shells is often preferred when revolving around the y-axis because it simplifies the integration.

Write and evaluate the definite integral that represents the volume of the solid formed by revolving the region about the y-axis for y = x^2.

Learn how to compute the volume of a solid formed by revolving the region under y = x^2 about the y-axis using the Cylindrical Shell Method. The solution includes step-by-step integration and the application of key calculus concepts, suitable for Grades 11-12 students.

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