find the volume of the solid obtained by rotating the region bounded by the curves y=x^2, x=y^2 about the line x=-2

To find the volume of the solid obtained by rotating the region bounded by the curves \( y = x^2 \) and \( x = y^2 \) about the line \( x = -2 \), we will use the **method of cylindrical shells**. This method is convenient when rotating around a vertical line.

### Step-by-Step Solution:

1. **Identify the curves and the region**:
   - The curve \( y = x^2 \) represents a parabola that opens upwards.
   - The curve \( x = y^2 \) represents a parabola that opens to the right.
   - To find the intersection points, solve the system of equations:
     \[
     x = y^2 \quad \text{and} \quad y = x^2.
     \]
     Substituting \( x = y^2 \) into \( y = x^2 \) gives \( y = (y^2)^2 = y^4 \), or \( y^4 - y = 0 \). Factoring gives:
     \[
     y(y^3 - 1) = 0 \quad \Rightarrow \quad y = 0 \quad \text{or} \quad y = 1.
     \]
     So, the curves intersect at the points \( (0, 0) \) and \( (1, 1) \).

2. **Set up the integral using the method of cylindrical shells**:
   We will integrate with respect to \( y \), from \( y = 0 \) to \( y = 1 \).
   The radius of a cylindrical shell at height \( y \) is the distance from \( x = -2 \) (the axis of rotation) to the curve \( x = y^2 \). The radius is:
   \[
   \text{Radius} = y^2 - (-2) = y^2 + 2.
   \]
   The height of the shell at height \( y \) is the horizontal distance between the curves \( x = y^2 \) and \( x = y^2 \):
   \[
   \text{Height} = y^2 - x^2.
   \]
   The formula for the volume of a cylindrical shell is:
   \[
   V = \.


Find the volume of the solid obtained by rotating the region bounded by the curves y=x^2, x=y^2 about the line x=-2.

Discover how to calculate the volume of the solid formed by rotating the region bounded by the curves y=x^2 and x=y^2 around the line x=-2 using the method of cylindrical shells. This detailed step-by-step solution involves curve intersections, cylindrical shells, and calculus-based techniques. Suitable for college-level calculus students.

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