Easy divergence theorem problem

[xmflea]

Evaluate the flux integral using the Divergence Theorem if F(x,y,z)=2xi+3yj+4zk
and S is the sphere x^2+y^2+z^2=9
answer is 324pi

so far i took the partial derivitavs of i j k for x y z and added them to get 9.

so i have the triple integral of 9 dzdxdy

i think u have to use polar cordinates, but I am reeally bad at those. and do not know how to set the bounds for this problem. can someone help me get started with that? thanks.

 

[xmflea]

see what i could do...is since i know volume of a sphere is 4/3r^3pi, i can just say that the radius is 3, and get 36pi x 9. to get 324pi. but I am sure my teacher would expect me to do some actual integration.

 

[HallsofIvy]

If I were your teacher, I wouldn't! Intelligence trumps blindly applying formulas any day.

If you really do want to integrate, use spherical coordinates. You book probably gives an example of using spherical coordinates to find the volume of a sphere. The "differential of volume" in spherical coordinates is $\rho^2 sin(\phi) d\rho d\theta\dphi$ and the limits of integration are $\rho$ from 0 to 3, $\theta$ from 0 to $2\pi$ and $\phi$ from 0 to $\phi$.