2 variable delta function integration

[Gin]

Homework Statement

$$\int^{A}_{-A}$$$$\int^{Bx}_{-Bx}c\delta(xcos\varphi+ysin\varphi-d)dydx$$
where A, B, c, d are constant

Homework Equations

The Attempt at a Solution

I have tried a few different ways to integrate this, but am completely confused with what happens to this kind of delta function when you integrate it. I know integrating a delta function usually gives you 1 but I don't think this can work in this case. The answer has A,B,c and d in it, so the limits must be used somewhere. This is one step in a much longer problem, but it is frustrating to get close to the end and get stuck because I can't find anything anywhere about delta functions of 2 variables. Some help would really be appreciated.

 

[Judithku]

Did you or anyone else figure out how to deal with this? I have the same problem.

 

[Mute]

One could, for example, use the following property of a delta function:

$$\delta(af(x)) = \frac{1}{|a|}\delta(f(x))$$

to factor out the cosine in the argument of the delta function and then perform the x integration. The x integration is then easy, but there's a trick - you don't know for sure if the delta function argument is zero inside the limits of x integration, so you'll have to think carefully about that.