# Resistance of a wire with changing radius

1. Jun 23, 2011

### brainslush

1. The problem statement, all variables and given/known data
The radius r of a wire of length L increases according to r = a * exp(bx^2), x is the distance from one end to the other end of the wire. What is the resistance of the wire?

2. Relevant equations
$R =\frac{L * \rho}{A}$

3. The attempt at a solution
$dR =\frac{dx * \rho}{A}$

$A(r) = \pi * r^2$

$r(x) = a* e^{bx^2}$

$A(x) =\pi a^2 * e^{2bx^2}$

$R =\int^{L}_{0}\frac{\rho}{\pi a^2 * e^{2bx^2}}*dx$

Two questions
First of all. Is this approach correct? And second, how does one integrate an errorfunction? (Of course one can use WolframAlpha but how does one get this solution?)
Thanks

Last edited: Jun 23, 2011
2. Jun 23, 2011

### L-x

Remember that A is a function of r when you try to work out dR

3. Jun 23, 2011

### brainslush

Of course, A is a function of r but r is also a function of x. Therefore the substitution yields what I have written above. Or am I wrong?

4. Jun 23, 2011

### L-x

$$R =\frac{x \rho}{A}=\frac{x \rho}{\pi r^2}=\frac{x \rho}{\pi a^2 e^{2bx^2}}$$

Are you sure your expression for dR is correct?

5. Jun 23, 2011

### brainslush

Our professor just told us that r = a * exp(b*x) and i found the mistake... A is dependent on r and therefore I need to consider this in my calculations...

$dR = \rho d(\frac{L}{A})$

$d(\frac{L}{A}) = \frac{dL}{A} - \frac{L*dA}{A^2}$

$dA = 2*\pi*r*dr$ , $dL = dx$

$dr = a*b*e^{bx} dx$

$dR =\rho \int^{L}_{0} (\frac{e^{-2bx}}{\pi*a} - \frac{2*x*b*e^{-2*b*x}}{\pi*a^2})dx$

Last edited: Jun 23, 2011