A Question Related to Fubini's Theorem

A Question Related to Fubini's Theorem

I was wondering if there is a integrable function $f: A \times B \to
\mathbb{R}$ (where $A, B \subset \mathbb{R}^n$ are rectangles) such that
$$ \int\limits_{A \times B} f = \int\limits_{A}\int\limits_{B} f(x, y) \,
dy \, dx $$ but $$ \int\limits_{B}\int\limits_{A} f(x, y) \, dx \, dy $$
does not exist.