Transform 2-dimensional integral to 1-dimensional one

Question

Simplify $\int_0^1 \int_0 ^x F(x,y)\,dy\,dx$ to a 1-dimensional integral. It is given that $F$ can be written in terms of $F:(x,y)\to f(y)$ with an integrable function $f:\mathbb R \to \mathbb R$. How does one handle this one?

Answer 1

Hint: use that $$\int_0^1\!\!\int_0^x F(x,y)\,dy\,dx = \int_0^1\!\!\int_0^x f(y)\,dy\,dx$$ and apply Fubini: $$ \int_0^1\!\!\int_0^x f(y)\,dy\,dx = \int_0^1\!\!\int_y^1 f(y)\,dx\,dy = \int_0^1(1 - y)f(y)\,dy . $$