TITLE: How to change order of integration in a double integral?
QUESTION [13 upvotes]: How would you go about changing the order of integration in a function say ; $$\int_0^8\int_\sqrt[3]{y}^2 f(x,y)~dx~dy$$

REPLY [19 votes]: \begin{eqnarray*}
\int_{0}^{8}\int_{\sqrt[3]{\vphantom{\large a}y\,}}^{2}{\rm f}\left(x, y\right)\,{\rm d}x\,{\rm d}y
& = &
\int_{0}^{8}\left\lbrack\int_{0}^{2}\Theta\left(x - \sqrt[3]{\vphantom{\large a}y\,} \right)
{\rm f}\left(x, y\right)\,{\rm d}x\right\rbrack{\rm d}y
\\
& = &
\int_{0}^{2}\left\lbrack\int_{0}^{8}\Theta\left(x^{3} - y\right)
{\rm f}\left(x, y\right)\,{\rm d}y\right\rbrack{\rm d}x
\\
& = &
\int_{0}^{2}\left\lbrack\int_{0}^{x^{3}}
{\rm f}\left(x, y\right)\,{\rm d}y\right\rbrack{\rm d}x
\end{eqnarray*}
where Θ is the Heaviside step function.