both iterated integrals exist and are equal, but the double integral does not

Consider the function $ f(x, y) = e^{ - x \cdot y } \sin x \sin y $for $ x \geq 0 $ and $ y \geq 0 $, and $ 0 $ in other cases.

Prove that both iterated integrals $$ \int_{\mathbb{R}} \left[ \int_{\mathbb{R}} f(x, y) \, dx \right] dy \quad \text{and} \quad \int_{\mathbb{R}} \left[ \int_{\mathbb{R}} f(x, y) \, dy \right] dx $$exist and are equal, but the double integral of $ f $ over $ \mathbb{R}^2 $ does not exist.

I have done the first part of the problem which proves both iterated integrals exist and are equal. I'm stuck at the second part showing the double integral of $ f $ over $ \mathbb{R}^2 $ does not exist, i.e $\iint_{\mathbb{R} \times \mathbb{R}} f(x,y) dx dy$. I found this link but it does not answer my question. Could someone explain why the double integral does not exist? Thanks in advance!