20.2 Precise Definition and Riemann Sums

The standard approach deals with a more general class of functions by imagining that we divide the interval between a and b into a large number of small strips and estimate the area of each strip to be the product of its width and some value of f(x) for x within the strip.

The area will then be something like the sum of the areas of the strips. If we then let the maximum strip length go to zero, we can hope to find the resulting sum of strip areas approach the true area.

The standard way to do this is to let the i-th strip begin at x_i-1 and end at x_i; the area of that strip is estimated as (x_i-x_i-1)f(x'_i) with x'_i anywhere in the strip.

A Riemann sum is a sum of the form just indicated: it is a sum over strips of the width of the strip times a value of the f(x) within the strip. The function is said to be Riemann integrable if the sum of the area of the strips approaches a constant independent of which arguments are used within each strip to estimate the area of the strip, as maximum strip width goes to zero.

The notation used above can be understood from this approach; we are summing the area of the individual strips, which for a very small interval around x of size dx is estimated to be f(x)dx, and summing this over all such strips. The integral sign represents the sum which is not an ordinary sum, but the limit of ordinary sums as the size of the intervals goes to zero.