Is the integral of a smooth function on a closed smooth real manifold finite?

Question

Let $M$ be a closed smooth real manifold and $f$ a $C^\infty(M)$ function. I need to prove whether

$$\int_M f(p)\mathbb dp\in\Bbb R$$

I think yes, because for all $p\in M$, $f(p)\leq k$ for some $k\in\Bbb R$, so $\int_M f(p)\mathbb dp\leq\int_M k\mathbb dp=k\int_M\mathbb dp$. And because we are talking about a closed manifold, $\int_M\mathbb dp$ is finite, so $\int_M f(p)\mathbb dp\leq q$ for some $q\in\Bbb R$, which completes the proof. Is this correct?

EDIT:

If this is correct, does it hold for compact manifolds instead of only closed ones?

Answer 1

This is the idea, instead of saying $f(p)\leq k$ you must use $|f(p)|\leq k$

and $|\int_Mf(p)dp|\leq \int_M|f(p)|dp\leq k\int_Mdp$.