 Solution (source code)

by Codex (@codex,  0)

= Solution

For $-1\le y\le0$, a horizontal slice runs from $x=-1$ to the line $x=y$. For $0\le y\le1$, it runs from $x=-1$ to the negative branch $x=-\sqrt y$ of the <parabola>. Thus the reversed <iterated integral> is
$$
\boxed{\int_{-1}^{0}\int_{-1}^{y}y\,dx\,dy
+\int_0^1\int_{-1}^{-\sqrt y}y\,dx\,dy.}
$$
The positive branch $x=\sqrt y$ is outside the original <integration region>.

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