For two closed manifolds $M$ and $N$ with dimension $m$ and $n$ respectively.
Note: If there is a differential map $f: M\to N$, then for any cohomology class $\beta\in H^*(N)$, $f^*\beta \in H^*(M)$.
Q: Under what kind of condition, the following formula $$\int_{M}f^*\beta=\int_{f(M)}\beta$$ always holds.
In general, a differential $k$-form can only be integrated over a $k$-manifold. So, in order for your equation to even make sense, $M$ and $N$ must have the same dimension (as if $\beta$ is an $n$-form so too will be $f^*\beta$).
For manifolds of the same dimension, this equation will hold iff $f$ is a degree-1 map (in this sense), more or less by definition.