The two term Weyl's Law states that $$N(\lambda)\sim\frac{area(\Omega)}{4\pi}\lambda-\frac{perimeter(\partial\Omega)}{4\pi}\sqrt\lambda$$ where $\Omega$ is a bounded domain in $R^2$, and $N(\lambda)$ is the number of eigenvalues (with multiplicity) $\leq\lambda$ in the Dirichlet eigenvalue problem for the Laplacian.
I know we can get the leading term using min-max theorems. How would I go about finding the second term, at least for some easy domains like rectangles.