Suppose that we want to calculate the following integral: $$\iint_{s+r<c}\lambda\exp(-\lambda s)\alpha\exp(-\alpha r)\delta(t-s-r)d(s,r)$$ where $\delta(.)$ is the delta dirac function and $t$ and $c$ are constants where $t<c$. When I do the integration first over $r$ and then over $s$, I get: $$\int_0^c\int_0^{c-s}\lambda\exp(-\lambda s)\alpha\exp(-\alpha r)\delta(t-s-r)drds$$ $$=\int_0^c\lambda\exp(-\lambda s)\alpha\exp(-\alpha(t-s))ds$$ $$=\alpha\lambda\exp(-\alpha t)\int_0^c\exp(-(\lambda-\alpha)s)ds$$ $$\frac{\alpha\lambda}{\lambda-\alpha}exp(-\alpha t)[1-\exp(-(\lambda-\alpha)c)]$$
But, when I change the order of integrations, I get a different result: $$\int_0^c\int_0^{c-r}\lambda\exp(-\lambda s)\alpha\exp(-\alpha r)\delta(t-s-r)dsdr$$ $$=\int_0^c\lambda\exp(-\lambda(t-r))\alpha\exp(-\alpha r)dr$$ $$=\alpha\lambda\exp(-\lambda t)\int_0^c\exp(-(\alpha-\lambda)r)dr$$ $$\frac{\alpha\lambda}{\lambda-\alpha}\exp(-\lambda t)[1-\exp(-(\alpha-\lambda)c)]$$
Is there any problem with my calculations?
You must consider carefully the case $t-s<0$: $$ \int_0^{c-s}e^{-\alpha r}\delta(t-s-r)dr=e^{-\alpha (t-s)}\theta(t-s), $$ where $\theta$ is Heaviside function.