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The Riemann Sum Formula For the Definite Integral

Part of 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:

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

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

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

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

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    The value w is the width of each rectangle:

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    Each h value is the height of a different rectangle:

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

    image7.png

  4. For a better approximation, use the limit

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

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