Find the volume of the solid that is generated by rotating around the indicated axis: y=x^2, y=8-x^2 the line y= -1

Use the washer method. Refer to the attached image below, which shows one such washer used to approximate the volume.

The volume of this kind of washer is , where is the radius of the larger cylinder and is the radius of the smaller cylinder. The height is the same for both and is given by , an infinitesimal change in . The radius of the larger cylinder is , while the radius of the smaller cylinder is just .

The two parabolas intersect at , so the volume of this solid is given by

Another method is to use Pappus's theorem, which states that if a region with area A is rotated about a line L, the volume of revolution generated is equal to
V=2(pi)RA, where R=distance of the centroid of region from the axis of rotation.

Here, 

Centroid (from symmetry) passes through y=4.
Volume = 2 π (4+1)(64/3) = 640 π /3 (approximately =670.2 ) units ³