In Paul Nahin's book, Dr. Euler's Fabulous Formula Cures Many Mathematical Ills, he proves the irrationality of $\pi^2$ using the differentiation operator $\mathbf{D}$, its inverse $\mathbf{D}^{-1}$ and Euler's formula.
On page 98, he writes about $\mathbf{D}^{-2}$, letting $f(x) = \mathbf{D}^{-1} \phi(x) = \displaystyle \int_{0}^{x} \phi(s)ds$,
$$\mathbf{D}^{-2}\phi(x) = \mathbf{D}^{-1}\mathbf{D}^{-1}\phi(x) = \mathbf{D}^{-1}f(x) = \displaystyle \int_{0}^{x}\displaystyle \left\{\int_{0}^{t}\phi(s)ds\right\}dt$$
He explains how the region for integration is the following:
And so, he justifies that the double integral can be rewritten as,
$$\mathbf{D}^{-2}\phi(x) = \displaystyle \int_{0}^{x}\displaystyle \left\{\int_{s}^{x}\phi(s)dt\right\}ds = \displaystyle \int_{0}^{x} \phi(s)ds \displaystyle \int_{s}^{x} dt = \displaystyle \int_{0}^{x}\phi(s)(x-s)ds$$
I can't satisfactorily follow his explanation of the region of integration, and hence, the equivalence of the double and single integrals. Can somebody please explain it?
Note The pdf version of the book is available here.
