I'm going to prove convergence of this application :
$$\phi \in D(\mathbb{R^{2}})$$
$$\langle T,\phi \rangle=\iint_{y>x^{2}+x}x\phi (x,y)\, dx\,dy$$
My try: $\phi \in D(\mathbb{R^{2}})$, so there exists $r>0$ such that $\operatorname{supp}\phi \subset\mathrm C(O,r)\subset\mathbb{R^{2}}$, where $\mathrm C$ is a circle.
Then: $r>x>-r$ and $x+x^{2}<y<r$, so:
$$\begin{split} \langle T,\phi \rangle &=\int_{[-r,r]}\int_{[x+x^2,r]}x\phi (x,y)\ dydx \\ &≤\sup_{\mathrm C} \int_{[-r,r]}\int_{[x+x^2,r]}x\ dydx \\ &=\sup_{\mathrm C} \int_{[-r,r]}|(rx-x^{2}-x^{3})|\ dx \\ &=r^{2}\sup_{\mathrm C}<\infty \end{split}$$
Is my solution correct?