For which $a$ does $f(x)=\frac{\arctan(x)}{\sqrt{x^a+x^{a+2}}}$ have finite volume?

Question

For which $a$ does $$f(x)=\frac{\arctan(x)}{\sqrt{x^a+x^{a+2}}}$$ have finite volume when rotated around x axis if $f:(0, \infty) \to \mathbb{R}$?

I already tried converting integral to the form: $$\pi \int_0^\infty \frac{\frac{\arctan^2(x)}{1+x^2}}{x^a}$$ so according to my textbook that should converge when $a>1$, but that is not correct.

Answer 1

The volume is $\pi\int_0^\infty f^2dx$. For small $x$, $f^2\sim x^{2-a}$, so we require $a<3$. For large $x$, $f^2\sim\frac{\pi^2}{4}x^{-a-2}$, so we require $a>-1$.