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For some quick background, when you use the areas of rectangles to estimate the area under a curve, that estimate is a Riemann Sum. When using this method you have two options: a Left Riemann Sum or a Right Riemann Sum. But what's the difference and when should you use each one?

The Difference:

  • Right Riemann Sum: The right corner of each rectangle touches the curve.
  • Increasing Function: Overestimate
  • Decreasing Function: Underestimate
  • Left Riemann Sum: The left corner of each rectangle touches the curve.
  • Increasing Function: Underestimate
  • Decreasing Function: Overestimate

How to Determine Which One to Use:

In all honesty, there are no set restrictions that determine when to use each one. Neither method is strictly better than the other.  It is up to you to look at how the rectangles fit within the curve to determine whether a Right or Left Riemann Sum will provide you with a more accurate estimate, but for the most part, the AP exam will tell you which approximation to use.

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