What is wrong with this computation of $\int_0^1\int_{-y}^y \sqrt[3]{x} \, dx \, dy$?
I'm considering real functions only. Since $x^{4/3}$ is an antiderivative of the integrand, we will get $\frac{3}{4}[x^{4/3}]_{-y}^y =\frac{3}{4}(y^{4/3}-(-y)^{4/3})=\frac{3}{4}(y^{4/3}-y^{4/3})=0$. Thus $\int_0^1 \int_{-y}^y \sqrt[3]{x} \, dx \, dy=0$. However, maple is giving me a complex (nonzero) number as the answer. Why is that? Any hint?
You are correct: $\sqrt[3]{x}$ is an odd function, so $\int_{-y}^y\sqrt[3]{x}\,dx$ must be zero.
The reason Maple is giving a nonsensical answer is because, viewed as a complex function, $x^{1/3}$ is multivalued, and they used a branch of $x^{1/3}$ which is positive real when $x$ is, and complex when $x$ is negative. It turns out this is the natural choice in complex analysis, but it is of course not what you would use in real calculus.
Here is a visualization of Maple's interpretation of $\sqrt[3]{x}$. Note that when $x<0$, $x^{1/3}$ has an imaginary part, which gives an imaginary contribution when this function is integrated.
It would be better to say, that antiderivative is $(\sqrt[3]{x})^4$, but in the rest you are right.
As others have already stated, Maple is interpreting $\sqrt[3]{x}$ as a complex function. Here's how it's being calculated $$ \int_0^1\int_{-y}^y\sqrt[3]{x}\ dxdy= \int_0^1\int_{-y}^y x^{\frac{1}{3}}\ dxdy $$ $$ = \int_0^1\left[\frac{x^{\frac{1}{3}+1}}{\frac{1}{3}+1}\right]_{-y}^y dy = \int_0^1\left[\frac{x^{\frac{4}{3}}}{\frac{4}{3}}\right]_{-y}^y dy $$ $$ =\frac{3}{4}\int_0^1\left[x^{\frac{4}{3}}\right]_{-y}^y dy=\frac{3}{4}\int_0^1 y^{\frac{4}{3}}-(-y)^{\frac{4}{3}} dy $$ $$ = \frac{3}{4}\left(\int_0^1 y^{\frac{4}{3}}dy-\int_0^1 (-y)^{\frac{4}{3}} dy\right) $$ Let $u=-y$, then $-du=dy$. So now we have $$ \frac{3}{4}\left(\int_0^1 y^{\frac{4}{3}}dy+\int_0^{-1} u^{\frac{4}{3}} du\right) $$ $$ = \frac{3}{4}\left(\left[\frac{y^{\frac{4}{3}+1}}{\frac{4}{3}+1}\right]_0^1 +\left[\frac{u^{\frac{4}{3}+1}}{\frac{4}{3}+1}\right] _0^{-1} \right) $$ $$ = \frac{3}{4}\left(\left[\frac{y^{\frac{7}{3}}}{\frac{7}{3}}\right]_0^1 +\left[\frac{u^{\frac{7}{3}}}{\frac{7}{3}}\right] _0^{-1} \right) $$ $$ = \frac{3^2}{4\cdot 7}\left(\left[y^{\frac{7}{3}}\right]_0^1 +\left[u^{\frac{7}{3}}\right] _0^{-1} \right) $$ $$ = \frac{9}{28}\left(1-0 +(-1)^{\frac{7}{3}}-0\right) $$ $$ = \frac{9}{28}\left(1+(-1)^{\frac{7}{3}}\right) = \frac{9}{28}\left(1+(-1)^{\frac{6}{3}+\frac{1}{3}}\right) $$ $$ =\frac{9}{28}\left(1+(-1)^{\frac{6}{3}}(-1)^{\frac{1}{3}}\right)= \frac{9}{28}\left(1+(-1)^{2}\sqrt[3]{-1}\right) $$ $$ =\frac{9}{28}\left(1+\sqrt[3]{-1}\right) $$
