The metric of the $3$-sphere is given by the line element $$ds^2=\frac{a^2}{1-r^2}dr^2+a^2r^2d\theta^2 + a^2r^2\sin^2\theta d\phi^2$$ Find the volume of the region $r\leq r_0$.
I am calculating the volume of the integral using the formula $$\int_0^{r_0}\int_0^\pi \int_0^{2\pi} \frac{a^3r^2\sin\theta}{\sqrt{1-r^2}}d\phi d\theta dr$$
Is this integral correct? I'm somehow not getting the correct answer. Are the limits correct?