# Correct use of substitution rule for Integration on Riemannian manifolds

Let $(N,g_N)$ be a Riemannian manifold and let $\psi: M \rightarrow N$ be a a diffeomorphism. Now I know how the Riemannian metric on $M$ defined by the pull-back of the metric on $N$ looks like (this is given in my case):

$$\psi^{\ast}g_N = \sum g_{ij} dx_i \otimes dx_j$$ is a Riemannian metric on $M$.

Now I want to compute Integrals of the form $$\int_M f \mathrm{dvol_{g_M}}$$ where $f \in C^{\infty}(M)$. But since I only have Information about $N$ and the pull-back I'd like to use the substitution rule.

In my case I know that $N=(0,\varepsilon) \times S^1$ and $\psi^{\ast}g_N = \frac{dx^2+d\theta^2}{x^2}$ for a small $\varepsilon > 0$ where $d\theta$ refers to the standard metric of $S^1$. How do I use the substitution rule for Integration correctly in my situation? Is it then correct to write $$\int_M f \mathrm{dvol_{g_M}} = \int_{\psi(M)} f(x,\theta) \frac{1}{x^2} dxd\theta$$ ?

As far as I know the substitution rule reads like $$\int_M (f \circ \psi) (x) \sqrt{\det D_\psi} \mathrm{dx} = \int_{\psi(M)} f(y) \mathrm{dy}$$ where $D_\psi$ is the Jacobian of the diffeomorphism $\psi: M \rightarrow N$.
More generally: If $w$ is a form on $N$ then: $$\int_M \psi^{\ast}w = \int_{\psi(M)} w$$
In my case the metric is given and defined by a pull-back like described. I am confused if I got it right: If the metric is defined by pullback, then the volume form is also a pull-back of a form on $N$? So in my case the corresponding form on $N$ is $\frac{\mathrm{dx\wedge d\theta}}{x^2}$? Is this correct with the above definitions? Did I compute the volume form correctly?