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General Riemann Sums

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

For the sake of the pictures, use a non-negative f. We split up the interval [a,b] into any n sub-intervals we like, not necessarily 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*1x1 + f (x*2x2 + ... + f (x*ixi + ... + f (x*{n – 1}x{ n – 1 } + f (x*nxn.

We can write this more tidily with summation notation as

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