Question

A nonconducting spherical shell, with an inner radius of 4 cm and an outer radius of 6 cm, has charge spread non uniformly through its volume between its inner and outer surfaces. The volume charge density ρ is the charge per unit volume, with the unit coulomb per cubic meter. For this shell ρ = b/r where r is the distance in meters from the center of the shell and b = 3 μ C/m2. What is the net charge in the shell?

Answer 1

In other words a infinitesimal segment dV caries the charge
dQ = ρ dV

Let dV be a spherical shell between between r and (r + dr):
dV = (4π/3)·( (r + dr)² - r³ )
= (4π/3)·( r³ + 3·r²·dr + 3·r·(dr)² + /dr)³ - r³ )
= (4π/3)·( 3·r²·dr + 3·r·(dr)² + /dr)³ )
drop higher order terms
= 4·π·r²·dr

To get total charge integrate over the whole volume of your object, i.e.
from ri to ra:
Q = ∫ dQ = ∫ ρ dV
= ∫ri→ra { (b/r)·4·π·r² } dr
= ∫ri→ra { 4·π·b·r } dr
= 2·π·b·( ra² - ri² )

With given parameters:
Q = 2·π · 3µC/m²·( (6cm)² - (4cm)² )
= 2·π · 3×10⁻⁶C/m²·( (6×10⁻²m)² - (4×10⁻²m)² )
= 3.77×10⁻⁸C
= 37.7nC