Flux of Vector Field within Sphere: Find Flux of Given Vector Field

[-EquinoX-]

Homework Statement

Suppose $$\vec{G}$$ is a vector field with the property that $$div\vec{G} = 5$$ for $$2 \leq ||\vec{r}|| \leq 14$$ and that the flux of $$\vec{G}$$ through the sphere of radius 4 centered at the origin is $$20\pi$$. Find the flux of through the sphere of radius 12 centered at the origin.

Homework Equations

The Attempt at a Solution

what I tried so far is

$$20\pi \int_0^{2\pi} \int_0^{\pi} \int_0^{12} \rho^2 sin(\phi)d\rho d\phi d\theta$$

is this wrong

 

[dx]

No integration is needed for this problem. The flux through the region 2 < R < 4 is just

5(volume of region 2 < R < 4)

by the divergence theorem. Use this, together with the fact that the flux through 0 < R < 4 is 20π, to find the flux through the region 0 < R < 2.

Can you see how to take it from here?

 

[-EquinoX-]

ok so after I got the flux through region 2< r< 4and through 0 < r< 4 I just substract it right?

 

[dx]

Yes, but in which order are you going to do the subtraction?

 

[-EquinoX-]

it's the 0<r<4 - 2<r<4 correct?

 

[dx]

Correct.