Prove that $\lim\limits_{r \to\infty} \int_1^{r-1} \int_1^{r-y} \frac{\sin x}{x^4+y^4} dx dy$ is finite

Question

Prove that $\lim\limits_{r \to\infty} \int_1^{r-1} \int_1^{r-y} \frac{\sin x}{x^4+y^4} dx dy$ exists (and is finite).

I was able to show that $\int_1^{r-y} \frac{\sin x}{x^4+y^4}dxdy$ converges when $r\to\infty$, but that didn't help. I also tried to use the fact that $|\int_{1}^{r-1}\frac{\sin x}{x^4+y^4}dxdy|\leq \int_1^{r-1} \int_1^{r-y} \frac{dxdy}{x^4+y^4}$, but that doesn't say anything as far as I understand.

So I basically have no clue. Hint, please?

Answer 1

HINT: The AM-GM inequality gives

$$x^4+y^4\ge 2x^2y^2$$