The Riemann integral is the classic integral you’re introduced to in introductory calculus classes; Normally shortened to just “integral,” it is found by taking the limit of Riemann sums.
Riemann Integral Overview
The idea behind Riemann integration is that you can find the integral of a bounded, real-valued function by finding the area of small rectangles close to the curve. If the rectangles are below the curve, it’s called the lower sum. Above the curve, it’s called the upper sum.
As these rectangles get smaller and smaller, they approach a limit. This limit will be a very good approximation for the integral, represented by the area under the curve. More specifically, the integral of a function f on interval [a, b] is a real number that can be interpreted geometrically as the signed (±) area under the graph y = f(x) for a ≤ x ≤ b (Hunter, n.d.).
Advantages and Disadvantages of the Riemann Integral
The Riemann integral is relatively simple to define and understand. With Riemann integrals, any continuous function (and many discontinuous functions) can be integrated.
Where Riemann integrals fail is in finding integrals for badly discontinuous or unbounded functions; In those cases, the Riemann integral simply doesn’t exist. This doesn’t mean that the integral doesn’t exist: even the most pathological of functions can be integrated using other methods like Lebesgue integration.
Next: Norm of a partition (i.e. the width of a Riemann sum rectangle).
References
Ferreirós, J. “The Riemann Integral.” §5.1.2 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkhäuser, pp. 150-153, 1999.
Hunter, J. Riemann Integral. Retrieved January 14, 2019 from: https://www.math.ucdavis.edu/~hunter/m125b/ch1.pdf
Jeffreys, H. and Jeffreys, B. S. “Integration: Riemann, Stieltjes.” §1.10 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 26-36, 1988.
Kestelman, H. “Riemann Integration.” Ch. 2 in Modern Theories of Integration, 2nd ed. New York: Dover, pp. 33-66, 1960.