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General Riemann Sums

A general Riemann sum is what we get when we estimate  using rectangles and not necessarily finding the rectangle on each sub-interval the same way.

For the sake of the pictures, we'll use a non-negative f. We split up the interval [a, b] into any n sub-intervals we like, not necessarily of the same size:

On each sub-interval, we pick a special x. It might be the left endpoint of the sub-interval, or the right-endpoint, or the midpoint, or somewhere else, and we don't need to pick the special x the same way on each interval. For each sub-interval, we use the function value at the special x as the height of the rectangle:

To write this in symbols instead of pictures, we use the counter i to keep track of which sub-interval we're on. The counter i will run from 1 to n, since there are n sub-intervals. Sub-interval i has width Δx_i and special x value x^*_i. This means the rectangle on sub-interval i has width Δx_i and height f (x^*_i):

Adding up the areas of all the rectangles, our general Riemann sum estimates the area between f and the x axis to be

f (x^*₁)Δx₁ + f (x^*₂)Δx₂ + ... + f (x^*_i)Δx_i + ... + f (x^*_{{n – 1}})Δx_{n – 1} + f (x^*_n)Δx_n.

We can write this more tidily with summation notation as