L2 space, function of 2 variables

Does anyone know how to show if a function of 2 variables is in L-2 Space (i.e. square integrable). I have only ever done this for functions of one variable and am not sure how the theory transfers over....

i have been asked to show that the function $u(r,\theta)=r^{2/3} sin((2\theta + \pi) /3)$ defined on the pi shaped domain $0\leq r \leq 1$ , $-\pi /2 \leq \theta \leq \pi$ is in $H^1(\Omega)$ but not in $H^2(\Omega)$. where the H's are Sobolev spaces. So i think i need to show that the first derivatives of u are square integrable but the second derivatives are not. This is where I became stuck, as the first and second derivatives are functions of two variables...

If you know how I could go about showing this I would be really gratefull, have been trying for ages I have been given the hint that I need to work in polar coordinates and use $dxdy=rdrd\theta$...

Thanks for your help, i think i know how to compute the double integrals.. are you implying that to show whether the functions of two variables (i.e. the first and second derivatives of u) are square integrable i need to integrate their square with respect to dxdy and show that this is finite???