Question

Use cylindrical coordinates. evaluate x dv e , where e is enclosed by the planes z = 0 and z = x + y + 10 and by the cylinders x2 + y2 = 16 and x2 + y2 = 36.

Answer 1

In cylindrical coordinates,

$\begin{cases}x(X,Y,Z)=X\cos Y\\y(X,Y,Z)=X\sin Y\\z(X,Y,Z)=Z\end{cases}\implies\mathrm dV=\mathrm dx\,\mathrm dy\,\mathrm dz=X\,\mathrm dX\,\mathrm dY\,\mathrm dZ$

Then the integral is

$\displaystyle\iiint_Ex\,\mathrm dV=\int_{Y=0}^{Y=2\pi}\int_{X=4}^{X=6}\int_{Z=0}^{Z=X\cos Y+X\sin Y+10}X\cos Y\times X\,\mathrm dZ\,\mathrm dX\,\mathrm dY$
$=\displaystyle\int_{Y=0}^{Y=2\pi}\int_{X=4}^{X=6}(10X^2\cos Y+X^3\cos^2Y+X^3\cos Y\sin Y)\,\mathrm dX\,\mathrm dY$
$=\displaystyle\int_{Y=0}^{Y=2\pi}\left(\dfrac{1520}3\cos Y+260\cos^2Y+260\cos Y\sin Y\right)\,\mathrm dY=260\pi$