Changing the order of integration without sketching?

$\Theta$ here refers to the Heaviside Step Function (wiki: http://en.wikipedia.org/wiki/Heaviside_step_function). It is defined very simply as:

$$\Theta(x) = \left\{\begin{array}{lr} 1 & x \geq 0 \\ 0 & x < 0 \end{array}\right.$$

The usefulness of $\Theta$ is that it allows one to express 'cutoffs' using function notation, and you can use this to change the bounds of integrals without changing their values.

So, for instance, you could write:

$$\int_1^{10}f(x)\,dx = \int_0^{10}\Theta(x - 1)f(x)\,dx$$

The reason they are the same is that $\Theta(x-1)$ is equal to zero on the interval from 0 to 1.

So this explains the first line of the manipulation:

$$\int_{\sqrt[3]{y}}^{2}f(x,y)\, dx = \int_0^{2}\Theta(x - \sqrt[3]{y})f(x,y)\, dx$$

The second line of the manipulation is Fubini's theorem, along with the observation that $x - \sqrt[3]{y} \geq 0$ if and only if $x^3 - y \geq 0$, so $\Theta(x - \sqrt[3]{y}) = \Theta(x^3 - y)$.

Note that you can express TWO cutoffs with two instances of the theta function:

$$\int_0^{1}\int_{y^2}^{y}f(x,y)\,dx\,dy = \int_0^1\int_0^1 \Theta(x - y^2) \Theta(y - x)f(x,y)\, dx\,dy$$