Let R be the region bounded by y=x^2, x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the line y=-2.

To find the volume of the solid generated by revolving the region R about the line y = -2 using the shell method, we can follow these steps:

1. Draw a graph of the region R bounded by the curves y = x^2, x = 1, and y = 0.

2. The axis of revolution is the line y = -2. Visualize the rotation of the region R around this line.

3. Now, imagine slicing the solid into infinitesimally thin vertical strips, or "shells," parallel to the y-axis. Each shell will be a hollow cylinder with thickness Δy and height h.

4. For a given value of y, the radius of each shell can be considered as the perpendicular distance from the line y = -2 to the corresponding x-value on the curve y = x^2. Let's denote this distance as r.

5. Since the axis of revolution is y = -2, the height h of each shell will be equal to y + 2.

6. The differential volume of each shell can be given by dV = 2πrhΔy, where 2π is the circumference of the cylinder.

7. We need to express r and h in terms of y.
- From the equation y = x^2, we can solve for x in terms of y.
x = sqrt(y)
- The perpendicular distance r from the line y = -2 to the curve y = x^2 is the difference between -2 and x.
r = -2 - sqrt(y)
- The height h of the shell is y + 2.
h = y + 2

8. Now, substitute the expressions for r and h into the differential volume formula: dV = 2π(-2 - sqrt(y))(y + 2)Δy.

9. Integrate the differential volume over the range of y from 0 to 1 (the limits of integration for the region R): V = ∫[0,1] 2π(-2 - sqrt(y))(y + 2)dy.

10. Evaluate this integral to find the volume V of the solid generated when region R is revolved about the line y = -2.