Order of integration for $\int_{0}^{T-N}\left\{ \int_{b}^{b+N}f(b,t)dt\right\} db$.

Question

I have the following integral

$$\int_{0}^{T-N}\left\{ \int_{b}^{b+N}f(b,t)dt\right\} db$$

How can I change the order of integration. $T$ and $N$ are constant variables. I tried to make a graphic which gives a paralleogram shape but after I could not understand how to change the order in a correct way.

Answer 1

The bounds of integration are not an issue for the Fubini-Tonelli theorem since $$ \int_{0}^{T-N}\left\{ \int_{b}^{b+N}f(b,t)dt\right\} db=\int_{\mathbb{R}^{2}}f(b,t)\cdot \boldsymbol{1}_{[b,b+N]}(t) \cdot \boldsymbol{1}_{[0,T-N]}(b)\,d(t,b). $$

Answer 2

First substitute $t=s+b$

$$\int_{0}^{T-N}\left( \int_{b}^{b+N}f(b,t)~\mathsf dt\right)\mathsf db=\int_0^{T-N}\left(\int_0^N f(b,s+b)~\mathsf d s\right)\mathsf d b$$

Answer 3

Nothing new, just wanted to illustrate how you can figure that out yourself in the future.