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To find the volume of the solid obtained by rotating the given region about the line \( x = -7 \), we need to apply the **method of cylindrical shells**.

### Step-by-step approach:

The region is bounded by the curves:

\[
x = -3 + y^2 \quad \text{and} \quad x = 2y
\]

These functions represent the left and right boundaries, respectively, in terms of \( y \). The limits of integration will be the \( y \)-values where these two curves intersect, which can be found by setting the two equations equal to each other:

\[
-3 + y^2 = 2y
\]

Solve this equation for \( y \).

Once the bounds are determined, the formula for the volume using the method of cylindrical shells is:

\[
V = 2\pi \int_{a}^{b} \left( r_{\text{shell}}(y) \right) h(y) \, dy
\]

- \( r_{\text{shell}}(y) \) is the distance from the shell to the axis of rotation \( x = -7 \). This is the difference between the function values and \(-7\).
- \( h(y) \) is the height of each shell, given by the difference between the two curves \( 2y \) and \( -3 + y^2 \).

Let me compute the details for the equation solving and the integral.The volume of the solid obtained by rotating the region bounded by \( x = -3 + y^2 \) and \( x = 2y \) about the line \( x = -7 \) is:

\[
V = 192\pi
\]

In numerical terms, this is approximately:

\[
V \approx 603.185789 \, \text{cubic units}
\]

Let me know if you'd like further details or explanations!

Here are 5 questions that expand on this problem:
1. How would the volume change if the axis of rotation was \( x = -3 \) instead of \( x = -7 \)?
2. Can we use the disk/washer method to solve this problem instead of cylindrical shells?
3. How would the integral change if the curves were rotated around the \( y \)-axis instead?
4. What would be the effect of rotating about a horizontal axis, like \( y = 0 \)?
5. How can we generalize this method to rotate regions about lines other than \( x = -7 \)?

**Tip:** Always sketch the region and the axis of rotation to better visualize the solid before applying any integration method!

Find the volume of the solid obtained by rotating the region bounded by x = -3 + y^2 and x = 2y about the line x = -7. Round to the nearest thousandth.

This problem involves finding the volume of a solid obtained by rotating the region bounded by x = -3 + y^2 and x = 2y around the line x = -7 using the method of cylindrical shells. The solution covers detailed steps for setting up and evaluating the integral.

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