I have a triple integral of this kind
$$\int_0^t{dx f(x)\int_{t-x}^{\infty}{dy g(y)\int_{t-x-y}^{t+\Delta t-x-y}{dz\delta(z)h(x,y,z)}}}$$
where $\delta$ is the Dirac Delta function and the other functions are well-behaved.
Since the integrand function is non-zero only for $z=0$, my idea was to locate the subdomain where $z=0$ (namely $[0,t]\times[t-x,t-x+\Delta t]$) and to rewrite the integral as
$$\int_0^t{dx f(x)\int_{t-x}^{t-x+\Delta t}{dy g(y)h(x,y,0)}}$$
The question is: is it correct or am I missing something?
I think the rigorous way would be to find a suitable change of coordinates, but I have no idea of how to do it.
Thank you very much, an answer to this question would help a lot!