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The problem shown in your image involves using the shell method to find the volume generated by revolving a shaded region around the y-axis. The shaded region is bounded by the curve \( y = x^2 \), the line \( x = 2 \), and the x-axis (i.e., \( y = 0 \)).

To solve this using the shell method, we follow these steps:

### Shell Method Formula
For rotation around the y-axis, the shell method formula for volume is:

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

where:
- \( x \) is the radius of the shell.
- \( f(x) \) is the height of the shell (i.e., the function that gives the curve’s height at \( x \)).
- The limits of integration, \( a \) and \( b \), are the bounds of the region in terms of \( x \).

### Setting Up the Integral
1. The function for the curve is \( f(x) = x^2 \), and we are revolving the region about the y-axis.
2. The limits of integration are from \( x = 0 \) to \( x = 2 \) (the vertical line bounding the region on the right).
3. The radius of each shell is \( x \), and the height is \( f(x) = x^2 \).

Thus, the volume integral is:

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

### Solving the Integral
Now we compute the integral:

\[
V = 2\pi \int_{0}^{2} x^3 \, dx = 2\pi \left[ \frac{x^4}{4} \right]_0^2
\]

\[
V = 2\pi \left( \frac{2^4}{4} - \frac{0^4}{4} \right) = 2\pi \left( \frac{16}{4} \right) = 2\pi \cdot 4 = 8\pi
\]

### Final Answer
The volume of the solid generated by revolving the shaded region about the y-axis is \( 8\pi \) cubic units.

Would you like more details on this solution or have any further questions? Here are some related problems to consider:

1. How would the solution change if the region was revolved around the x-axis instead of the y-axis?
2. What happens if the curve was \( y = x^3 \) instead of \( y = x^2 \)?
3. How do you apply the disk/washer method for the same problem?
4. What is the difference between the shell method and disk/washer method in general?
5. Can you derive the volume if the curve was bounded by \( y = 4 - x^2 \) instead?

**Tip**: When using the shell method, always identify the radius and height of the cylindrical shells carefully based on the axis of rotation.

Use the shell method to find the volume generated by revolving the shaded region about the y-axis. The region is bounded by the curve y = x^2, the line x = 2, and the x-axis (y = 0).

Learn how to use the shell method to calculate the volume generated by revolving the region bounded by y = x^2, x = 2, and y = 0 around the y-axis. This problem covers essential calculus concepts, including volume of solids of revolution and integration. Suitable for high school seniors and college students.

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