The Riemann Integral

Abstract

As was pointed out in the previous chapter, the second fundamental topic covered in calculus is the Riemann Integral, the first being the derivative. For a (nonnegative) real-valued function of a real variable, this integral extends the notion of area, defined initially for rectangles: For a nonnegative constant function f(x) := c ∀x ∈ [a, b], the area of the rectangle bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b, is defined to be the non-negative number A := (b - a)c. This is then trivially extended to the case of step functions which are piecewise constant Simply add the areas of the finite number of rectangles involved. This suggests the following analytic approach to the general case: Try to approximate the given function by step functions, find the areas corresponding to the latter functions as above, and pass to the limit. Our objective in this chapter is to provide a mathematically rigorous foundation for this intuitive approach. We begin with some basic definitions.