Compute the volume of the solid formed by revolving the fourth quadrant region bounded by y = x^2 - 1 , y = 0, and x = 0 about the line y = 6.

Answer 1

If you use discs, each disc has a hole in it or radius 6. So, the volume is

Int(pi*(R^2-r^2) dx)[0,1]
= Int(pi*(7-y)^2 - 7^2)dx)[0,1]
= 148/15 pi

If you use shells, each shell has height x, thickness dy.
x = (y+1)^1/2

Int(2pi*(7-y)*x dy)[-1,0]
2pi*Int(7-y)*sqrt(y+1) dy)[-1,0]
= 148/15 pi