Determining Charge from Charge Density

[Sunbodi]

Homework Statement

A positively charged nonconducting solid sphere of radius R has a nonuniform volume charge density given by ρ0 for r≤R/2 and given by 2ρ0(1−r/R) for R/2 ≤r≤R, where r is the radial distance from the sphere center.Part A
Determine the charge q on the sphere in terms of ρ0 and R.

Homework Equations

∫ρ*dv = Q₀∫^R/2 ρ0 dVolume + _R/2∫^R ρ0 dVolume = Q from 0 to R/2 + Q from R/2 to R

The Attempt at a Solution

₀∫^R/2 ρ0 dVolume = ₀∫^R/2 ρ0 4πr² dr
= R³πρ0/2_R/2∫^R ρ0 dVolume = _R/2∫^R ρ0 4πr² dr

= (4R³- 3R⁴)*8πρ0/(12R)

Is my math wrong or is my attempt at the solution wrong? If both is wrong could you help me more with the attempt at the solution?

Thanks!

Sun

(ρ) Is rho btw.

 

[haruspex]

What is ∫r².dr?

In the second integral, you left out the (1-r/R) factor.