Please tell me if what I did is correct or if there's any faster alternatives.
I set $x$ and $y$ axes on the center of the circle with radius $r$, therefore this can be seen as an area described by $x^2+y^2=r$ revolving around $x=-R-r$
$dV$ can be written as $dV = 2\pi(R+x)\cdot 2y \cdot dx =2\pi(R+x)\cdot 2 \sqrt{r-x^2} \cdot dx$
$V$ is then
$ \int_R^{R+2r} 4\pi(R+x)\sqrt{r-x^2} dx $
Is this correct?