Past exam of the mathematical methods II course of the University of Oxford / 2021 / 5

Interpret the cube root as the real cube root. The curved boundary is $y=-x^3$, so the bounded region is $0\le x\le 3$, $-x^3\le y\le 0$. In particular $y$ is negative inside it. The double integral is the convergent improper integralFor negative real $y$, an antiderivative is $(3/2)y^{2/3}=(3/2)|y{|}^{2/3}$. The inner integral is therefore $-3x^2/2$. HenceThe negative sign is essential. Absolute convergence follows by the same calculation with the sign removed, so the singular edge at $y=0$ causes no ambiguity.