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 Δxi and special x value x*i. This means the rectangle on sub-interval i has width Δxi 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)Δxi + ... + f (x*{n – 1})Δxn – 1 + f (x*n)Δxn.
We can write this more tidily with summation notation as
