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Estimating Area as Sum of Rectangles

7.1 Estimating Area as Sum of Rectangles

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:
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