# Definite Integrals

# I Like Abstract Stuff; Why Should I Care?

## 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 Δ*x _{i}* and special

*x*value

*x*

^{*}

_{i}. This means the rectangle on sub-interval

*i*has width Δ

*x*and height

_{i}*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})Δ*x*_{1} + *f* (*x*^{*}_{2})Δ*x*_{2} + ... + *f* (*x*^{*}_{i})Δ*x*_{i} + ... + *f* (*x*^{*}_{{n – 1}})Δ*x*_{{ n – 1 }} + *f* (*x*^{*}_{n})Δ*x _{n}*.

We can write this more tidily with summation notation as