Question

Find the volume of the given solid region in the first octant bounded by the plane 9z+15y+15z=45 and the coordinate​ planes, using triple integrals.

Answer 1

First,

$9x+15y+15z=45\implies 3x+5y+5z=15$

The volume is given by the integral (one of 6 possible combinations),

$\displaystyle\int_0^5\int_0^{\frac{15-3x}5}\int_0^{\frac{15-3x-5y}5}\mathrm dz\,\mathrm dy\,\mathrm dx=\boxed{\frac{15}2}$