I'm struggling with the following problem. I have some $Y_\alpha = \alpha(A)$ parameterized manifold in $\mathbb{R}^n$ with $A \subset \mathbb{R}^k$ open, $k \leq n$, and $\alpha \in C^r(A)$. Given an isometry (no further assumptions given) $h \colon \mathbb{R}^n \to \mathbb{R}^n$ and $Z_\beta = h(Y_\alpha)$ for $\beta = h \circ \alpha$, show that $Y_\alpha, Z_\beta$ have the same volume such that$$\int_AV(D\alpha) = \int_AV(D\beta).$$
My initial instinct was to either
(a) apply chain rule $D\beta = D(h(\alpha)) \circ D\alpha$ and somehow use that $|Dh| = 1$ over $A$, or
(b) somehow use the equivalence relation given by $Y_\alpha \sim Y_\beta \iff \exists g: A \to B$ invertible in $C^r(A)$ such that $\alpha = \beta \circ g$ with the fact that $Y_\alpha \sim Y_\beta \implies V(Y_\alpha) = V(Y_\beta)$ and somehow relate this back to $Z_\beta$.
I have not learned anything about isometries, and only little about parameterized manifolds thus far, so this question is really stumping me. I would appreciate any type of hints or content about isometries that I am missing (no spoiling the solution).