Finding total charge from volume charge density

[emr564]

Why do I have to integrate? Isn't it redundant?

Homework Statement

The volume charge density inside a solid sphere of radius a is given by ρ=ρnaught*r/a, where ρnaught is a constant. Find the total charge as a function of distance r from the center.

Homework Equations

Q=ρV

The Attempt at a Solution

Q=(ρnaught*r/a)*(4/3)(∏a^3)
Q=(4/3)∏ρnaught*r*a^2

but the solution manual says it's supposed to be Q=∏ρnaught*a^3
It says that you have to integrate ρ with respect to V, and that's what confuses me. If you integrate, you're taking the charge of one tiny sphere and adding it the the charge of a concentric sphere a little bigger, so on and so forth, so doesn't that mean that each time you move up to a bigger sphere, you're being redundant? Why can't you multiply the total volume by the charge per volume to get charge?

 

[Spinnor]

The charge density is zero at the center and gets larger with increasing r, as you wrote,

ρ=ρnaught*r/a

 

[emr564]

Thank you! I finally noticed that after way too long...