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The Fundamental Theorem of Calculus Lesson 7.4

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2 Definite Integral Recall that the definite integral was defined as But … finding the limit is not often convenient We need a better way!

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3 Fundamental Theorem of Calculus Given function f(x), continuous on [a, b] Let F(x) be any antiderivative of f Then we claim that The definite integral is equal to the difference of the two antiderivatives Shazzam !

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4 What About + C ? The constant C was needed for the indefinite integral It is not needed for the definite integral The C's cancel out by subtraction

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5 You Gotta Try It Consider What is F(x), the antiderivative? Evaluate F(5) – F(0)

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6 Properties Bringing out a constant factor The integral of a sum is the sum of the integrals

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7 Properties Splitting an integral f must be continuous on interval containing a, b, and c

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8 Example Consider Which property is being used to find F(x), the antiderivative? Evaluate F(4) – F(0)

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9 Try Another What is Hint … combine to get a single power of x What is F(x)? What is F(3) – F(1)?

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10 When Substitution Is Used Consider u = 4m 3 + 2 du = 12m 2 It is best to change the new limits to be in terms of u When m = 0, u = 2 When m = 3, u = 110

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11 Area Why would this definite integral give a negative area? The f(x) values are negative You must take this into account if you want the area between the axis and the curve

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12 Assignment Lesson 7.4A Page 399 Exercises 1 – 43 odd Lesson 7.4B Page 401 Exercises 53 – 65 odd

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