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Definite Integrals Finding areas using the Fundamental Theorem of Calculus

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The Definite Integral Definition: The definite integral of f(x) from x=a to x=b is Really, the definite integral computes the area under the curve by adding up the area of an ‘infinite’ number of rectangles

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Riemann Sums Become The Definite Integral Increase the number of rectangle to get closer to the area under the curve

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Computing Definite Integrals One approach is to compute a left or right Riemann sum for large numbers (~100) of rectangles. Computing Riemann Sums

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A Better Way Of course, you wouldn’t want to do this a lot (unless you have to) The Fundamental Theorem of Calculus says that if you want to compute find a function F(x) so that F 0 (x) = f(x). Then

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Using the FTC Any antiderivative F(x) will do so pick the one with C=0 Ex: Evaluate

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Another Example Evaluate s 0 1 e 2x dx. Solution: First, find an antiderivative of e 2x. This is F(x) = (1/2)e 2x (why?). Now compute that s 0 1 e 2x dx = F(1)-F(0) = (1/2)e 2 – (1/2)e 0 = e 2 /2-1/2 = 3.1945…

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