Question

Evaluate the triple integral. 10x dv, where e is bounded by the paraboloid x = 2y2 + 2z2 and the plane x = 2. e

Answer 1

Convert to a rotated set of cylindrical coordinates, taking

$x=t$

$y=r\cos\theta$

$z=r\cos\theta$

As in standard cylindrical coordinates, the volume element is the same

$\mathrm dV=\mathrm dx\,\mathrm dy\,\mathrm dz=r\,\mathrm dr\,\mathrm d\theta\,\mathrm dt$

Then the integral is

$\displaystyle\iiint_E10x\,\mathrm dV=10\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=2}\int_{t=2r^2}^{t=2}rt\,\mathrm dt\,\mathrm dr\,\mathrm d\theta=\frac{1040\pi}3$