First let me write out the definition of a manifold given in my book:

Let $\displaystyle k > 0 $. A k-manifold in $\displaystyle \mathbb{R}^n $ of class $\displaystyle C^r $ is a subspace $\displaystyle M $ of $\displaystyle \mathbb{R}^n $ having the following property: For each $\displaystyle p \in M $, there is an open set $\displaystyle V \subset M $ containing $\displaystyle p $, a set $\displaystyle U $ that is open in either $\displaystyle \mathbb{R}^k $ or $\displaystyle \mathbb{H}^k $ (upper half space), and a continuous bijection $\displaystyle \alpha : U \rightarrow V $ such that 1) $\displaystyle \alpha $ is of class $\displaystyle C^r $, 2) $\displaystyle \alpha^{-1} : V \rightarrow U $ is continuous, 3) $\displaystyle D\alpha(x) $ has rank k for each $\displaystyle x \in U $. The map $\displaystyle \alpha $ is called a coordinate patch on $\displaystyle M $ about $\displaystyle p $.

In my textbook I am reading the chapter on integrating a scalar map over a compact manifold. My question is this: Suppose M is a compact-manifold. As a subset of $\displaystyle \mathbb{R}^n $ it is bounded. So instead of going through all the mess of defining a manifold and defining the integral of a continuous function f over a manifold, why not just integrate f over M as one usually would? Using Riemann sums in $\displaystyle \mathbb{R}^n $?

Surely this would give the same result as integrating f over M using the definition of integral over a manifold. So what's so special about using a manifold M for integration when we could just consider M as a regular bounded subset of Euclidean space and integrate it how we usually would?

In case you're wondering, here is the definition of the integral of a scalar map over a compact manifold M:

Let M be a compact k-manifold in $\displaystyle \mathbb{R}^n $, of class $\displaystyle C^r $. Let $\displaystyle f: M \rightarrow \mathbb{R} $ be a continuous function. Suppose that $\displaystyle \alpha_i: A_i \rightarrow M_i $, for i = 1, ..., N, is a coordinate patch on M, such that $\displaystyle A_i $ is open in $\displaystyle \mathbb{R}^k $ and M is the disjoint union of the open sets $\displaystyle M_1, M_2, ..., M_N $ of M and a set K of measure zero in M. Then $\displaystyle \int_M f dV = \sum_{i = 1}^N \int_{A_i} f(\alpha_i) V(D \alpha_i) $.

Note that $\displaystyle dV $ represents the integral with respect to volume and $\displaystyle V(D \alpha_i) = \sqrt{det[(D\alpha_i)^{tr} D\alpha_i]} $