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The problem asks to set up an integral to describe the volume of a solid obtained by rotating a region bounded by two curves around the axis \( y = -2 \).

### Step 1: Understand the curves
The given curves are:
1. \( x = 1 + (y - 2)^2 \)
2. \( x = 5 - y^2 \)

These curves are functions of \( y \), so we are working with a horizontal axis of rotation. The curves are to be rotated around the axis \( y = -2 \).

### Step 2: Identify bounds of integration
To set up the integral, we first find the points where the curves intersect, which will determine the bounds for \( y \). We need to solve for \( y \) where:
\[
1 + (y - 2)^2 = 5 - y^2
\]
Solving this equation will give the limits of integration.

### Step 3: Distance from the axis of rotation
The axis of rotation is \( y = -2 \). To use the method of cylindrical shells or washers, we need the distance from each curve to the axis of rotation. The distance for each curve is \( (y - (-2)) = y + 2 \).

### Step 4: Setting up the integral using the washer method
We use the washer method, where the volume of a solid of revolution is:
\[
V = \pi \int_{y_1}^{y_2} \left[ R_{\text{outer}}^2 - R_{\text{inner}}^2 \right] \, dy
\]
Where:
- \( R_{\text{outer}}(y) \) is the distance from the axis to the outer curve \( x = 5 - y^2 \),
- \( R_{\text{inner}}(y) \) is the distance from the axis to the inner curve \( x = 1 + (y - 2)^2 \),
- \( y_1 \) and \( y_2 \) are the bounds of integration, obtained from the intersection points.

### Step 5: Expressing the radii
- \( R_{\text{outer}}(y) = 5 - y^2 \)
- \( R_{\text{inner}}(y) = 1 + (y - 2)^2 \)

### Step 6: Final integral setup
The volume integral is:
\[
V = \pi \int_{y_1}^{y_2} \left[ (5 - y^2)^2 - (1 + (y - 2)^2)^2 \right] \, dy
\]
Where \( y_1 \) and \( y_2 \) are the values where the curves intersect.

Would you like me to solve for the bounds of integration or explain further steps?

### Related Questions:
1. How do you solve for the intersection points of the two curves?
2. Why is the washer method used instead of the disk method?
3. What are other methods of finding the volume of solids of revolution?
4. How does shifting the axis of rotation affect the integral setup?
5. How do you handle cases where the curves are not easily solvable for intersections?

#### Tip:
Always carefully determine the radius in washer or shell method problems. The radius depends on the distance from the axis of rotation, which may be offset from the standard \( y = 0 \).

Set up an integral to describe the volume of the solid obtained by rotating the region bound by the curves x = 1 + (y - 2)^2 and x = 5 - y^2 around the axis y = -2.

Learn how to set up an integral to find the volume of a solid formed by rotating the region bounded by the curves x = 1 + (y - 2)^2 and x = 5 - y^2 around the axis y = -2. This step-by-step solution covers key calculus concepts, including the washer method and calculating intersection points.

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