Find the volumes of the solids generated by revolving the regions about the given axes.
The region bounded by y= square root x y= (x^2)/8
a.) the x-axis
b.) the y-axis
points of intersection
$\displaystyle \sqrt{x} = \frac{x^2}{8}$
$\displaystyle x = 0$ is the obvious solution
$\displaystyle 8 = x^{\frac{3}{2}}$
$\displaystyle x = 4$
about the x-axis, use washers ...
$\displaystyle V = \pi \int_0^4 (\sqrt{x})^2 - \left(\frac{x^2}{8}\right)^2 \, dx$
... or cylindrical shells
$\displaystyle V = 2\pi \int_0^2 y(\sqrt{8y} - y^2) \, dy$
about the y-axis, use cylindrical shells ...
$\displaystyle V = 2\pi \int_0^4 x\left(\sqrt{x} - \frac{x^2}{8}\right) \, dx$
... or washers
$\displaystyle V = \pi \int_0^2 (\sqrt{8y})^2 - (y^2)^2 \, dy$