Choose a volume form $\omega$ on $M$, oriented manifold. For every $F\in C^{\infty}_c(M)$, we define $$ \int_M F:=\int_M F\omega $$ where in the right hand term $M$ is taken wit positive orientation w.r.t. $\omega$, that is $$ \omega=\lambda_p dx_1\wedge\dots\wedge dx_n $$ for every $p$, and $\lambda_p>0$.
Let $M$ be a $n$-dimensional compact manifold and let $\omega\in\Omega^n(M)$ be a volume form. Define the volume of M as $$ Vol(M):=\int_M 1=\int_M\omega $$
Now I'm asking myself if the Volume is well-defined, i.e. if it doesn't depend on the choice of $\omega$.
I would say that it depends on the choice of $\omega$.
Infact, I know that $\omega$ would be, locally, something of the kind $$ (\omega)_p=F_p(dx_1\wedge\dots\wedge dx_n)_p $$ where $x_1,\dots,x_n$ are the coordinates induced by a local chart and $F\in C^{\infty}(M)$.
So $\omega$ is not unique.
Probably I haven't learnt well this topic.
Remember that a volume form for a $n$-dimensional manifold $M$ is a line section of the canonical bundle $\Lambda^nT^*M$, i.e. a non-vanishing $n$-form.
How can I prove that the volume of a compact manifold always positive?