I am trying to find value of the following integral:
$\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} dx \int_{-1}^{1} dy\ \delta(\sin2x)\ \delta(x-y)$
I have some experience in solving integrals that contain Dirac delta function in one variable, but here it is in two variable. I am completely clueless and don't even know how to approach this problem. I am not expecting the solution, need someone to guide me what approach will help me to evaluate this integral?
To extend change-of-variables manipulations to the Dirac delta, the following composition is defined: $$\delta(\mathbf f(\mathbf x))=\sum_{\mathbf r_i}\frac{\delta(\mathbf x-\mathbf r_i)}{|\mathbf J_{\mathbf f}(\mathbf r_i)|}$$ where $\mathbf r_i$ are the roots of $\mathbf f$. For the given $\mathbf f(x,y)=(\sin2x,x-y)$, there is only one root in the domain of integration, $(0,0)$, and $$|\mathbf J_{\mathbf f}|=\begin{vmatrix}2\cos2x&1\\0&-1\end{vmatrix}=-2\cos2x$$ Evaluating this at $(0,0)$ and taking the absolute value gives 2, so (within the domain of integration) $$\delta(\mathbf f(\mathbf x))=\frac{\delta(\mathbf x-(0,0))}2$$ Thus the integral over the domain is $\frac12$.