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Properties of the integral of step functions

It's easy to prove the following geometric interpretation of the integral of a step function.

Given two numbers a and b and a step function s whose domain is [a,b] (if a<b), or [b,a] (if b<a), consider the integral img. Now draw the directed "path" from (a,0) to (b,0), to (b,s(b)), to
(a,s(a)) and again to (a,0), as in the following picture. This path encloses some regions, whose contour is sometimes clockwise and sometimes counterclockwise directed. We'll consider as positive the areas whose contour is counterclockwise directed, as negative those whose contour is clockwise directed. The integral img is the algebraic sum of these areas.

step functions and area

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Main properties

We now point out some fundamental properties of the integral of step functions. These properties are almost obvious if one keeps in mind the geometric interpretation and, for this reason, we leave the proof as an exercise. The most important fact is that all these properties remain unchanged also for the integral of more general functions, as we shall see.

  1. Linearity property. Given two step functions s and t and two real numbers c and d, the following holds: 
  2. Additive property with respect to the integration interval. Given three numbers a, b and c (whose order is unimportant), and a step function s, the following holds:
  3. Comparison property. Given two step functions s and t and two real numbers a and b (with a < b), if s(x)t(x) for all x in [a,b] then: img

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first published on january 07 2003 - last updated on september 01 2003