Volume of a sphere in Schwarzschild metric

[tb87]

Homework Statement

Calculate the volume of a sphere of radius $r$ in the Schwarzschild metric.

Homework Equations

I know that
\begin{align}
dV&=\sqrt{g_\text{11}g_\text{22}g_\text{33}}dx^1dx^2dx^3 \nonumber \\
&= \sqrt{(1-r_s/r)^{-1}(r^2)(r^2\sin^2\theta)} \nonumber
\end{align}
in the Schwarzschild metric.

The Attempt at a Solution

Well the integral I get for the sphere's volume,
\begin{equation}
V = \int dV \nonumber
\end{equation}
gives an imaginary volume! What's going on? Of course the volume will be imaginary because $dV$ is imaginary when $r<r_s$ (plus, there's a singularity at $r=r_s$, which complicates things if we want to integrate up to the Schwarzschild radius). There's obviously something I'm missing here, but I have no idea what it is.

 

[tb87]

Update : e-mailed my teacher and there's something we haven't time to see in class (Kruskal coordinates) that was required for this problem. -_-