What is the complexity of calculating the potential of a cylinder?

[Buffu]

Homework Statement

Homework Equations

The Attempt at a Solution

The position of the point (where V is to calculated) on the z-axis would be $u = z_0 + l/2$.So in cylindrical coords,

$$V(u) = \int_V {k \rho \over (s^2 + (u -z)^2)^{1/2}} dV = k \rho \int_0^L \int_0^{2\pi} \int_0^R {k \rho \over (s^2 + (u -z)^2)^{1/2} } \ ds\ d\phi\ dz \\= 2\pi k \rho \int_0^L (R^2 + (u -z)^2)^{1/2} - (u - z)\ dz =2\pi k \rho \left[\int_0^L (R^2 + (u-z)^2)^{1/2} dz - z_0L \right]$$

This integral is very complex and cubersome to calculate. I think I made a mistake some where.

 

[TSny]

Your work looks good to me. The last integral does turn out to yield a messy answer.