Approximate Integration: Simpson's Rule: Proof

In this video I go over another integral approximation method and this time I derive the Simpson's Rule method which uses parabolas to estimate the area under a curve. This method moves away from the traditional straight lines segments used in the Left, Right, Midpoint, and Trapezoidal approximation methods and has curved segments in the shape of parabolas. This method is named after the English Mathematician Thomas Simpson (who lived from 1710 to 1761) who popularized this method in his best-selling calculus textbook: A New Treatise of Fluxions. Note that fluxions are what Isaac Newton used to refer to derivatives but eventually Leibnitz derivative notation won out. Also, even though the Simpson's Rule was actually known to Cavalieri and Gregory, Simpson greatly popularized it! The proof shown in this video is very clever and combined with the brief mathematical history lesson, makes this a very useful video to watch!

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Related Videos:

Approximate Integration: Example 3: Midpoint Rule: https://youtu.be/SxHfJ3MrkSk Approximate Integration: Midpoint Rule Error Bound: Proof: http://youtu.be/aKUFHXNeW7Y Approximate Integration: Example 1: 1/x: http://youtu.be/DTzZ1jz6OOg Approximate Integration: Accuracy and Error Bounds: http://youtu.be/WTdNktL_1Uc Approximate Integration: Trapezoidal Rule Error Bound: Proof: http://youtu.be/_mrSHIin7Mw Integration by Parts: Integration Constants: http://youtu.be/jJ-RJTNMy4s Evaluating Integrals - Midpoint vs Right Endpoint Approximations Comparison: http://youtu.be/3x3sF7P9xfY .

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