Question

Find the volume of the region between the cylinder z equals y^2 and the? xy-plane that is bounded by the planes x equals 0 x=0?, x equals 4x=4?, y equals negative 4y=?4?, and y equals 4y=4.

Answer 1

The volume is given by the triple integral

$\displaystyle\iiint_V\mathrm dV=\int_{x=0}^{x=4}\int_{y=-4}^{y=4}\int_{z=0}^{z=y^2}\mathrm dz\,\mathrm dy\,\mathrm dx$

which evaluates to

$\displaystyle4\int_{y=-4}^{y=4}y^2\,\mathrm dy=\dfrac{512}3$

where the coefficient 4 comes from integrating with respect to $x$, and the integrand $y^2$ comes from integrating with respect to $z$.