Another prelim problem:
Suppose that $f(x,y)$ is a smooth function defined on $\mathbf{R}^2$. Prove that $$ \int_{x^2+4y^2\leq r^2}f(x,y)\,dx\,dy = ar^2+br^4+O(r^5) $$ Express $a$, and $b$ in terms of $f$ and other constants independent of $f$ and $r$.
I was thinking to integrate by parts, but it is in multidimensions. Is it possible to use Green's Theorem to make this an integral around the ellipsis so that it is in one variable and then integrate by parts?
Thanks!
Hint: Let $x=r\cos\theta$, $y=\frac{1}{2}r\sin\theta$ and note that $$f(x,y)=f(0,0)+x\frac{\partial f}{\partial x}(0,0)+y\frac{\partial f}{\partial y}(0,0)+\frac{1}{2}\left(x^2\frac{\partial^2 f}{\partial x^2}(0,0)+2xy\frac{\partial^2 f}{\partial x\partial y}(0,0)+y^2\frac{\partial^2 f}{\partial y^2}(0,0)\right)+O(r^3).$$