Math Help - Cylinder with uniform charge distribution - algebra block!

1. Cylinder with uniform charge distribution - algebra block!

I have a cylinder whose radius, r = a & length = L
charge distribution through cylinder is
ρ = C/2 * r3
where r = radial distance in cylindrical coordinates
C = constant
show that the average charge density ρbar = Ca3 / 5

* * * * * * * * * * * * *

I have average charge = tot. charge / tot. volume

aAGH! Amendment
vol =
∏r2L

i.e. ρ / ∏r2L

= ((C/2)*r3) / ∏r2L

= Cr3 / 2∏r2L

= Cr / 2∏L

= Cr2 / 6.28L

Now this should make more sense to me but doesn't.

2. Re: Cylinder with uniform charge distribution - algebra block!

$\rho$ is not the "tot. charge", it is the charge at a single point. It's been a few years since I took calculus (so maybe someone else can correct me if I'm mistaken), but I believe you need to calculate:
$\int_{r=0}^{r=a}{\int_{\theta=0}^{\theta=2\pi}{ \frac{C}{2} r^3 \,d\theta\,dr\ }} = 2\pi \int_0^a{ \frac{C}{2} r^3 \,dr }$

3. Re: Cylinder with uniform charge distribution - algebra block!

Of course.
All makes perfect sense.