I am not really familiar with the notion of principal part. I guess the following statement is true: (that I extracted it from sketch of a theorem)
Suppose $f=g+dh+dk$ on a smooth subset $U$ of $\Bbb R^n$ and principal part of $f$ and $dh+dk$ are equal to zero then $$\int_{B(p,r)} f\ \mathsf{dvol}\sim\int_{B(p,r)} g\ \mathsf{dvol}.$$ Where $p\in U$ and $B(p,r)$ is a disk of radius $r$ centered at $p$.
I am not sure about the exact statement. I would be grateful if someone could provide some references (suitable for differential geometers if possible) that contains the above statement and its proof and some solved problems.