Differential Equations For Dummies
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The Fundamental Theorem of Calculus has a shortcut version that makes finding the area under a curve a snap. Here it is. Let F be any antiderivative of the function f; then

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With this version of the Fundamental Theorem, you can easily compute a definite integral like

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You could get this area with two different methods that involve area functions. First, you could determine the area function for this parabola that begins sweeping out area at x = 2, and then compute that area function’s output when x = 3. Second, you could determine the area function for the parabola that begins sweeping out area at x = 0, and then use that area function to subtract the area from 0 to 2 from the area from 0 to 3.

The beauty of the shortcut theorem is that you don’t have to use an area function like

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or any other area function.

You just find any antiderivative, F(x), of your function, and do the subtraction, F(b) – F(a). The simplest antiderivative to use is the one where C = 0. So, here’s how you use the theorem to find the area under the parabola from 2 to 3.

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and thus,

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About This Article

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Steven Holzner was an award-winning author of more than 130 books, of which more than 2 million copies have been sold. His books have been translated into 23 languages. He served on the Physics faculty at Cornell University for more than a decade, teaching both Physics 101 and Physics 102. Holzner received his doctorate in physics from Cornell and performed his undergraduate work at Massachusetts Institute of Technology, where he also served as a faculty member.

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