Good evening!
During my calculation I meet such integral:
$$\int\limits_{0}^{1} dx \int\limits_{f(x)}^{g(x)} dy \,\delta(x - y)$$
I know how Dirac delta act on functions which we integrate, but I confused how it changes limits.
One my idea is that we will get this:
$$\int\limits_{0}^{1} (g(x) - f(x) )\, dx$$
Am I right?
$$\int_{f(x)}^{g(x)} \delta(x-y) dy = \begin{cases} 1 & f(x) \leq x \leq g(x) \\ -1 & g(x) \leq x \leq f(x) \\ 0 & \text{otherwise} \end{cases}.$$
Other than the $-1$ you are basically checking whether the delta is ever evaluated at zero. The $-1$ is there in case you are ever integrating with the limits in reverse order.
Note that the limits are not changed, we are just using the definition of the Dirac delta for each $x$.