Integrating over a Manifold

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

Let . A k-manifold in of class is a subspace of having the following property: For each , there is an open set containing , a set that is open in either or (upper half space), and a continuous bijection such that 1) is of class , 2) is continuous, 3) has rank k for each . The map is called a coordinate patch on about .

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 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 ?

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 , of class . Let be a continuous function. Suppose that , for i = 1, ..., N, is a coordinate patch on M, such that is open in and M is the disjoint union of the open sets of M and a set K of measure zero in M. Then .

Note that represents the integral with respect to volume and