Question

Evaluate the triple integral. 5x dv, where e is bounded by the paraboloid x = 7y2 + 7z2 and the plane x = 7. e

Answer 1

Convert to cylindrical coordinates:

$x=x$

$y=r\cos\theta$

$z=r\sin\theta$

Then $E$ is the set of points $(r,\theta,x)$ such that $0\le r\le1$, $0\le\theta\le2\pi$, and $7y^2+7z^2\le x\le7$ or $7r^2\le x\le7$.

Now

$\displaystyle\iiint_E5x\,\mathrm dV=5\int_0^{2\pi}\int_0^1\int_{7r^2}^7xr\,\mathrm dx\,\mathrm dr\,\mathrm d\theta=\boxed{\frac{245\pi}3}$