Use the method of cylindrical shells to find the volume v generated by rotating the region bounded by the given curves about y = -7. y=x^2, x=y^2.

Attached is a diagram setting up the problem.
Because the axis of rotation is horizontal and we are using shell method, the radius will be in the y-direction and integration will need to be in terms of "y".

The general formula for shell method is:


r(y) is the radius as shown in diagram and is equal to the distance from y=-7 to upper curve x=y^2. This is distance from y=-7 to x-axis plus y-value along curve.
---> r(y) = y+7

h(y) is height of shell represented as x-distance between the 2 curves.
y =x^2 --> x = sqrt(y),   x = y^2
----> h(y) = sqrt(y) - y^2

The limits are determined by y-values of intersection of 2 curves.
y^2 = sqrt(y) ---> y^3 -y = 0   -----> y = 0,1

Now we can write the integral: