Question
So I'm from an engineering and recently learned about integrating on riemannian manifolds. However, I have been faced with the notation $$\int _{T_xM} f(z) \ dz,$$ where $M$ is a Riemannian manifold and $T_x$ is the tangent space at some point $x$. Note that I have learned that $T_xM$ is the space of all tangent vectors at $x$. Can anyone say what this integration notation means? I cannot find any references online.
I have been taught that if $M$ is a Riemannian manifold then $$\int_M f=\int_{\mathbb{R}^n}f \sqrt{\mathrm{det}(A^tA)} \ dx_1\ldots,dx_n$$ where $A$ is the jacobian (I think).