When I want to know the volume of the Euclidean space $\mathbb{R}^3$, I have to calculate
$$ V = \int_0^a dx \int_0^bdy\int_0^cdz $$
$a,b,c$ are the maximum values of $x,y,z$
How about in the case of non-Euclidean space? Now I treat the space which is called $\bf{q}$-space in my research. ${\bf q} = (q_1, q_2, q_3)$
And I know the metric $g({\bf q})$ which is given as 3×3 matrix and the connection $\Gamma({\bf q})$ of this space. This space is flat. It means $\Gamma({\bf q}) = 0$. I want to know the volume of this space when $q_1 \in [0, \alpha]$, $q_2 \in [0, \beta]$, $q_3 \in [0, \gamma]$.
I think the volume of this space $V$ is not
$$ V = \int_0^{\alpha} dq_1 \int_0^{\beta} dq_2\int_0^{\gamma} dq_3 $$
I don't know your conventions or contexts, so the following explanation might be different from you want, but I hope this answer makes some help.
I guess your metric is given by a matrix form. I mean, for any point $\mathbf{q}$, $g(\mathbf{q})$ is a positive-definite symmetric matrix of size $3 \times 3$. In this situation, you can define the volume of $U$ as follows, where $U := \{\ (q_1,q_2,q_3) \ |\ q_1 \in [0,\alpha], q_2 \in [0,\beta] , q_3 \in [0,\gamma] \ \}$;
\begin{equation} \int_U \sqrt{|\det g(\mathbf{q})|} dq_1 dq_2 dq_3. \end{equation}
Under some mild conditions, this is equal to the iterated integral.
This is known as the volume form of Riemannian manifolds. This name might be a help for your further research :)
By the way, if you want to know/define the volume of Riemannian manifolds, you don't need connections. In other words, the volume does not depend on the connection.