very strange phenomenon $f(x,y)=x^4-6x^2y^2+y^4$ integral goes wild I am going over my lecture's notes in preparation for exam and I saw something a bit strange I would like someone to explain how it is possible.
Look at the function $f(x,y) = x^4-6x^2y^2+y^4$
if we convert it to polar coordinates, we will get $f(r,\theta)=r^4\cos(4\theta)$
the weird thing about this function, is that it has 2 different integrals over the same area.
What I mean: if we calculate the integral $$ \lim_{n \to \infty}\int_{0}^{2\pi} \int_{0}^{n} r^5\cos(4\theta)dr d\theta$$ we will see it is equal to zero.
However, if we calculate $$\lim_{n \to \infty}\int_{-n}^{n} \int_{-n}^{n} x^4-6x^2y^2+y^4dxdy = \lim_{n \to \infty}\int_{-n}^{n} \frac{2n^5}{5}-4n^3y^2+2ny^4 dy=\lim_{n \to \infty} -\frac{16}{15}n^6 = -\infty$$
How come the integral is so different based just on how we choose to represent the same area? (it is $\mathbb R^2$ in both cases. first it is a giant circle, second case it is a giant square)
So if asked what is $$\iint_{\mathbb R^2} x^4-6x^2y^2+y^4$$ is the correct thing to say that it does not exist?

The integral is unbounded when integrated over the entire plane of $\mathbb R^2$, what you are doing by changing the variables is analogous to changing the order of the terms in an non-converging summation, you can get more than one "answer" but in fact none of them are valid.
