A cylindrical conductor of resistivity ρ has length L. Its radius varies uniformly from r1 at one end to r2 at the other end (r2>r1).
(i) Consider a thin slice of thickness dx cut at a distance x from the end where the radius is r1. If the radius at this position is r(x), write an expression for the resistance dR of this slice.
(ii) Using the linear variation of the radius along the length, derive an expression for the total resistance R in terms of ρ, L, r1, and r2.
(iii) Show explicitly that when r1=r2, your expression reduces to the standard formula for the resistance of a uniform cylindrical wire.