What is the volume of the solid

the solid is generated when the region bounded by y = x^2, y = 1, and x = 0 is revolved about the line y = -1

The axis of rotation is horizontal, so the disk method will require us to slice vertically, implying dx. Note that that limits will extend from 0 to 1:

V = ∫ A(x) dx from 0 to 1

A(x) = πR^2 - πr^2 = π(R^2 - r^2)

So first determine the outer radius, R. The outer radius is defined by the line y = 1. Since the rotation axis is at y = -1 though, R is always 2 away, so R = 2. Next determine the inner radius, r. The inner radius is defined by y = x^2, but again, since the rotation axis is at y = -1,r = x^2 + 1. Sub those in:

A(x) = π(2^2 - (x^2 + 1)^2)
A(x) = π(-x^4 - 2x^2 + 3)

V = π ∫ [-x^4 - 2x^2 + 3] dx from 0 to 1
V = π[(-1/5)x^5 - (2/3)x^3 + 3x] from 0 to 1

Now evaluate and note that the lower limit of 0 will be 0:

V = π[(-1/5)(1)^5 - (2/3)(1)^3 + 3(1)]
V = (32/15)π

Done!