What is the difference between Riemann integration and Lebesgue integration?

Answer 1

Think of Lebesgue integration as an extension of Riemann integration to functions that are not as well-behaved. Riemann integration works when functions are relatively smooth in the sense that the area under the curve of a Riemann integrable function can be approximated by rectangles. When functions are not as well-behaved, we need a finer way to measure area (by finer, I mean better than using only rectangles).