I would need to determine whether the following double integral converges or diverges:
$$\int_1^\infty \int_0^x \frac{1}{x^3+y^3}\, dy\, dx$$
I made a change of variable to polar coordinates and therefore I got:
$$\int_0^\frac{\pi}{4} \int_1^\infty \frac{r}{r^3\cos^3(\theta)+r^3\sin^3(\theta)} \,dr \,d\theta$$
Which simplifies to:
$$\int_0^\frac{\pi}{4} \frac{1}{\cos^3(\theta)+\sin^3(\theta)} d\theta$$
I can't get any further from here. I cannot evaluate this integral by hand and I don't know what I should do next. Is this approach correct in this problem?
Thanks!
Best Answer
Another solution using no fancy change of variables. Just note that $\frac{1}{x^3+y^3}\leq \frac{1}{x^3}$ for $y\geq 0$ and thus,
$$ \int_1^{\infty} \int_0^x \frac{1}{x^3+y^3}\textrm{d}y\textrm{d}x\leq \int_1^{\infty} \frac{1}{x^2}\textrm{d}x<\infty $$