please solve this:

Question

we have a cylinder with radius a and length L and a conductivity of δ=δ"(r) find the resistivity of it please .

Answer 1

You need to split the cylinder into concentric cylindrical shells. All of these shells conduct current in parallel along the length of the cylinder.

Consider a shell at a distance r from the center of the cylinder and with thickness dr. Let the resistance of this shell be dR. Since these shells are in parallel, you need to use the formula for the parallel combination of resistors. If you have a finite number of resistors, then 1⁄Reff=1⁄R1+1/R2+...

In this case, you have an infinite number of shells. So you need to integrate 1/dR.

Lets first find dR

dR=L/(Aδ(r)) and 1/dR=Aδ(r)/L

The area of the shell can be thought of a thin strip of length 2πrdr

Therefore, 1/dR=Aδ(r)/L=2πrdrδ(r)/L

1/R_eff=∫1/dR=∫(2πrδ(r)/L)dr

If you integrate this from 0 to r you would get the answer to the total resistance.

I think the problem should say "find the resistance" not the resistivity. The resistivity of the cylinder is a property of the material of the cylinder and it would just be the reciprocal of the conductivity.