Evaluate the following using cylindrical coordinates:
$\displaystyle \int\int\int_E(x^3+xy^2)dV$
where E is the solid that lies in the first octant beneath $\displaystyle z=16-x^2-y^2$
Hi
$\displaystyle x^2+y^2 =16-z$ means that z varies from 0 to 16
$\displaystyle x^2+y^2 = r^2 = 16-z$ therefore for any value for z between 0 and 16, r varies from 0 to $\displaystyle \sqrt{16-z}$
The first octant means that $\displaystyle \theta$ varies from 0 to $\displaystyle \frac{\pi}{2}$
Then $\displaystyle x^3+xy^2 = r^3 \cos^3 \theta + r \cos \theta r^2 \sin^2 \theta = r^3 \cos \theta$
and $\displaystyle dx dy$ is to be replaced by $\displaystyle r dr d\theta$
$\displaystyle \int\int\int_E(x^3+xy^2)dV = \int_{0}^{16} \int_{0}^{\frac{\pi}{2}} \int_{0}^{\sqrt{16-z}} r^4 \cos \theta dr d\theta dz$