Estimating Area as Sum of Rectangles
- The area under a graph within an interval x \in [a,\ b] can be estimated by a sum of rectangles for smaller intervals of x.
- There are two types of constructing these rectangles, namely the left-endpoint estimate and right-endpoint estimate. Both estimates will probably lead to a different answer.
- There is no rule as left-endpoint estimate will yield a smaller or bigger area. It depends on the curve, but a general rule can be applied:
i) For f decreasing over [a,\ b]: left-endpoint estimate ≥ true area ≥ right-endpoint estimate
ii) For f increasing over [a,\ b]: left-endpoint estimate ≤ true area ≤ right-endpoint estimate
- The steps of estimating area under a curve as sum of rectangles are: