Question

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

Answer 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 $\pi R^2h-\pi r^2h$, where $R$ is the radius of the larger cylinder and $r$ is the radius of the smaller cylinder. The height $h$ is the same for both and is given by $\mathrm dx$, an infinitesimal change in $x$. The radius of the larger cylinder is $(8-2x^2)+(x^2+1)=9-x^2$, while the radius of the smaller cylinder is just $x^2+1$.

The two parabolas intersect at $x=\pm2$, so the volume of this solid is given by


$\displaystyle\pi\int_{x=-2}^{x=2}(9-x^2)^2-(x^2+1)^2\,\mathrm dx=\frac{640\pi}3$

Answer 2

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, 
$A=\int_{-2}^{2}(8-x^2-x^2)dx=64/3$
Centroid (from symmetry) passes through y=4.
Volume = 2 π (4+1)(64/3) = 640 π /3 (approximately =670.2 ) units ³