Riemann integral/Above and below/Section

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A step function from above and one from below. The green area is a step integral from below and the yellow area (partly covered) is a step function from above.

Let denote a compact interval and let

denote a function. Then is called Riemann-integrable if the upper integral and the lower integral

of exist and coincide.

It might by historically more adequate to call this Darboux-integrable.

Let denote a compact interval. For a Riemann-integrable function

we call the upper integral of (which by definition coincides with the lower integral) the definite integral of over . It is denoted by

The computation of such integrals is called to integrate. Don't think too much about the symbol . It expresses that we want to integrate with respect to this variable. The name of the variable is not relevant, we have

Let denote a compact interval, and let

denote a function. Suppose that there exists a sequence of lower step functions  with and a sequence of upper step functions  with . Suppose furthermore that the corresponding sequences of step integrals converge to the same real number. Then is Riemann-integrable, and the definite integral equals this limit, so


We consider the function

which is strictly increasing in this interval. Hence, for a subinterval , the value is the minimum, and is the maximum of the function on this subinterval. Let be a positive natural number. We partition the interval into the subintervals , , of length . The step integral for the corresponding lower step function is

(see exercise for the formula for the sum of the squares). Since the sequences and converge to , the limit for of these step integrals equals . The step integral for the corresponding step function from above is

The limit of this sequence is again . By fact, the upper integral and the lower integral coincide, hence the function is Riemann-integrable, and for the definite integral we get

Let be a compact interval, and let

be a function. Then the following statements are equivalent.
  1. The function is Riemann-integrable.
  2. There exists a partition , such that the restrictions are Riemann-integrable.
  3. For every partition , the restrictions are Riemann-integrable.
In this situation, the equation



Let be a function on a real interval. Then is called Riemann-integrable, if the restriction of to every compact interval is


Due to this lemma, both definitions coincide for a compact interval . The integrability of a function does not mean that has a meaning or exists.