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The Riemann Sum Formula For the Definite Integral

Part of the Calculus II For Dummies Cheat Sheet

The Riemann Sum formula provides a precise definition of the definite integral as the limit of an infinite series. The Riemann Sum formula is as follows:

image0.png

Below are the steps for approximating an integral using six rectangles:

image1.png
  1. Increase the number of rectangles (n) to create a better approximation:

    image2.png
  2. Simplify this formula by factoring out w from each term:

    image3.png
  3. Use the summation symbol to make this formula even more compact:

    image4.png

    The value w is the width of each rectangle:

    image5.png

    Each h value is the height of a different rectangle:

    image6.png

    So here is the Riemann Sum formula for approximating an integral using n rectangles:

    image7.png
  4. For a better approximation, use the limit

    image8.png
  5. to allow the number of rectangles to approach infinity:

    image9.png
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