Integrating Fresnel Functions (Double Integral)

[Eddie91]

Homework Statement

All that is provided can be found through the following link:

http://img33.imageshack.us/img33/6343/question2q.jpg

Homework Equations

No specific equations pertaining to solving double integrals.

The Attempt at a Solution

Ok, so I know that we cannot integrate this with elementary functions as it stands at the moment. Experience with these double integral questions tells me that I should change the order of integration.

Here's my first problem though, and that's figuring out exactly what the new bounds should be (I think I'm getting caught up in all the variables). I sketched the region on an x-t plane, which essentially consisted of a line passing through the origin x=t, and making the observation that x = alpha = t.

Just considering integrating the "cos" function first, here's what I came up with:

http://img31.imageshack.us/img31/9828/question2attempt.jpg

Am I on the right track? If not, how should I proceed?

I appreciate the help guys :)

 

[mighty2000]

Based on your attempt at a solution, it seems like you are on the right track. Your intuition about changing the order of integration is correct, as that is often a useful strategy in solving double integrals.

To determine the new bounds for integration, it may be helpful to think about the geometry of the region being integrated. In this case, the region is bounded by the lines x=0, x=t, and x=alpha=t. This forms a triangle in the x-t plane, with vertices at (0,0), (alpha,0), and (alpha,alpha).

To integrate with respect to x first, the bounds for x would be 0 to t, as you have correctly identified. To integrate with respect to t second, the bounds would be from 0 to alpha, as this represents the upper limit for t within the triangle region.

Your attempt at setting up the integral for the "cos" function looks correct. Just be sure to also include the limits of integration for the "sin" function, which would be from 0 to t for the inner integral and from 0 to alpha for the outer integral.

I hope this helps guide you towards a successful solution. Good luck with your problem!