Definition 5.3.1 (Volume of a manifold). Let $M \subset \R^n$ be a smooth $k$-dimensional manifold, $U$ a pavable set of $\R^k$, and $\gamma: U\to M$ be a parametrization according to definition 5.2.3. Let $X$ be as in that definition. Then

$$ \text{vol}kM = \int{\gamma(U-X)}|d^kx|\\ = \int_{U-X}(|d^kx|(P_{\gamma(u)}(\vec D_1\gamma(u), \dots, \vec D_k \gamma(u))))|d^ku|\\ = \int_{U-X}\sqrt{\det ([D\gamma(u)]^T[D\gamma(u)])}|d^ku|. $$

Definition 5.3.2 (Integrals over smooth manifolds, with respect to volume). Let $M\subset \R^n$ be a smooth $k$-dimensional manifold, $U$ a pavable set of $\R^k$, and $\gamma: U \to M$ a parametrization. Let $X$ be as in Definition 5.2.3. Then $f: M \to \R$ is integrable over $M$ with respect to volume if the integral on the right of equation 5.3.5 exists, and then

$$ \int_M f(x) |d^kx| = \int_{U-X}f(\gamma(u))\sqrt{\det ([D\gamma(u)]^T[D\gamma(u)])}|d^ku|. $$