|
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 ith strip begin at xi-1
and end at xi; the area of that strip is estimated as (xi-xi-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 rectangles, which for a very small interval around
x of size dx is estimated to be f(x)dx, and summing this over all such intervals.
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.
|
