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There are two tools, generalizing a concept of a volume to the case of submanifolds in $\mathbb{R}^n$, namely the Hausdorff measure $H^k$ and the volume form. The question is how to show that if $M$ is an orientable $k$-submanifold in $\mathbb{R}^n$ with a volume form $dV$ then $$ \int\limits_{M} f(x) dV = \int\limits_{M} f(x) H^k(dx) $$ if the integrals exist.

P.S. Maybe the question is too silly for MathOverflow and more suitable for Mathematics Stack Exchange, but I have 2 reasons to post it here:

1. In books on geometric integration theory (Krantz, Parks; Federer) I failed to find an answer.
2. I've already posted the question on Mathematics Stack Exchange and on one more forum, but I didn't receive any response.

If this question is very silly, I will delete it.

Thank you.

There are two tools, generalizing a concept of a volume to the case of submanifolds in $\mathbb{R}^n$, namely the Hausdorff measure $H^k$ and the volume form. The question is how to show that if $M$ is an orientable $k$-submanifold in $\mathbb{R}^n$ with a volume form $dV$ then $$ \int\limits_{M} f(x) dV = \int\limits_{M} f(x) H^k(dx) $$ if the integrals exist.

P.S. Maybe the question is too silly for MathOverflow and more suitable for Mathematics Stack Exchange, but I have 2 reasons to post it here:

1. In books on geometric integration theory (Krantz, Parks; Federer) I failed to find an answer.
2. I've already posted the question on Mathematics Stack Exchange and on one more forum, but I didn't receive any response.