In this lecture we introduce the notion of a smooth singular cube in a smooth manifold, and explain how to integrate a differential form over a singular cube. We then define a singular chain as a formal sum of singular cubes, and explain how the smooth singular cubes form a chain complex.

The homology of this chain complex can be related to the de Rham cohomology via the de Rham Theorem and Poincaré Duality. We will discuss this in the final lecture of the semester.