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."

For a bounded function on a finite closed interval, the definition requires a common limit of the Riemann sums as the mesh size tends to zero, independently of the chosen sample points (Hunter 2014, pp. 236-237).

The construction is consistent with Jordan measure. The one-dimensional definition and its counterparts on rectangles and cuboids take the forms

			(1)
			(2)
			(3)

where are finite. In one dimension, is an arbitrary point in the th subinterval and is the mesh size. For the multiple integrals, the sample point lies in the corresponding cell of area or volume , and is the maximum side length of all cells in the rectangular partition (Lebl 2026, pp. 97-98). Requiring only the maximum area or volume to tend to zero would allow cells to remain long in one direction. Convergence for one chosen sequence of partitions and sample points alone does not establish that the function has a Riemann integral.

As an example of the application of the Riemann integral definition, find the area under the curve from 0 to , where and . 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.