flux using divergence theorem

**TheRekz** · April 28th 2009, 06:04 PM

Let W be the solid bounded by the paraboloid $x = y^2 + z^2$ and the plane x = 9. Let $\vec{F} = 5x\vec{i} + y\vec{j} + z\vec{k}$ . I know the divF is 7 here.. but after that I will need to multiply this by the volume of the region bounded above to get the flux.. am I right?

**redsoxfan325** · April 28th 2009, 09:59 PM

Originally Posted by TheRekz

Let W be the solid bounded by the paraboloid $x = y^2 + z^2$ and the plane x = 9. Let $\vec{F} = 5x\vec{i} + y\vec{j} + z\vec{k}$ . I know that $div(\vec{F})=7$ here.. but after that I will need to multiply this by the volume of the region bounded above to get the flux.. am I right?

Yes, you want to integrate $\int\int\int 7\,dV$ .

I would use cylindrical coordinates (note that I'm using x where one would traditionally use z):
$x=x$
$r^2=y^2+z^2$
$dV=r\,dx\,dr\,d\theta$

So your volume is bounded by $x=r^2$ and $x=9$

$\theta=0..2\pi$
$r=0..3$
$x=r^2..9$

$7\int_0^{2\pi}\int_0^3\int_{r^2}^9 r\,dx\,dr\,d\theta$

Spoiler: