Electrical resistance of a paraboloid.

[BlackWyvern]

Homework Statement

What would the electrical resistance of a paraboloid from y = 0 to L be?

Homework Equations

$$R = \rho \frac{L}{A}$$

The Attempt at a Solution

Okay, so I'll put the parabola (that would rotate into the paraboloid) into the form $$y = \sqrt{x}$$

The function A(x) is just the area of the circle, at distance x.

A = $$\pi y^{2} = \pi x$$

I'll break the paraboloid up first into a finite sum of discs, from 0 to L.

$$R = \Sigma \rho \frac{\Delta x}{A(x)}$$

==>

$$R = \int^{L}_{0} \rho \frac{dx}{\pi x}$$

==>

$$R = \frac{\rho}{\pi} \int^{L}_{0}\frac{1}{x} dx$$

This integral resolves to:

$$R = [\frac{\rho}{\pi}ln(x)]^{L}_{0}$$

Natural log of 0 is undefined, so this would resolve to a numerical answer, as at the limit of x ==> 0, the area approaches zero and this means infinite resistance. But is the maths correct? Or can someone suggest a better way to actually find an answer.

 

[BlackWyvern]

bump. <_<