Riemann Integral

The Riemann integral is the definite integral normally encountered in calculus texts and used by physicists and engineers. Other types of integrals exist (e.g., the Lebesgue integral), but are unlikely to be encountered outside the confines of advanced mathematics texts. In fact, according to Jeffreys and Jeffreys (1988, p. 29), "it appears that cases where these methods [i.e., generalizations of the Riemann integral] are applicable and Riemann's [definition of the integral] is not are too rare in physics to repay the extra difficulty."

The Riemann integral is based on the Jordan measure, and defined by taking a limit of a Riemann sum,

			(1)
			(2)
			(3)

where  and , , and  are arbitrary points in the intervals , , and , respectively. The value  is called the mesh size of a partition of the interval  into subintervals .

As an example of the application of the Riemann integral definition, find the area under the curve  from 0 to . Divide  into  segments, so , then

			(4)
			(5)
			(6)

By induction

(7)

so

(8)

(9)

For example, take .

(10)

so

			(11)
			(12)
			(13)
			(14)

Riemann integrals can be computed only for proper integrals.

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.

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 rev. ed. New York: Dover, pp. 33-66, 1960.