If $M$ is a Riemannian manifold and $\phi$ is a Borel measurable function into $M$, then what is the relationship between $\int_M \phi \,d\mu$ and $\int_{\mathbb{R}^d} \psi\circ \phi\, d\lambda$; where $\mu$ is the Riemannian measure on $M$, $\lambda$ is the Lebesgue measure on ${\mathbb{R}^d}$ and $\psi: M \rightarrow {\mathbb{R}^d}$ maps $M-\mathrm{Cutlocus}(p)$ (for some $p \in M$) to its normal coordinates?
Specifically is $\int_{\mathbb{R}^d} \psi\circ \phi \,d\lambda\leq \int_M \phi\, d\mu$?