The Fundamental Theorems of Calculus Lesson 5.4. First Fundamental Theorem of Calculus Given f is  continuous on interval [a, b]  F is any function.

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Presentation transcript:

The Fundamental Theorems of Calculus Lesson 5.4

First Fundamental Theorem of Calculus Given f is  continuous on interval [a, b]  F is any function that satisfies F’(x) = f(x) Then

First Fundamental Theorem of Calculus The definite integral can be computed by  finding an antiderivative F on interval [a,b]  evaluating at limits a and b and subtracting Try

Area Under a Curve Consider Area =

Area Under a Curve Find the area under the following function on the interval [1, 4]

Second Fundamental Theorem of Calculus Often useful to think of the following form We can consider this to be a function in terms of x View QuickTime Movie View QuickTime Movie

Second Fundamental Theorem of Calculus Suppose we are given G(x) What is G’(x)?

Second Fundamental Theorem of Calculus Note that Then What about ? Since this is a constant …

Second Fundamental Theorem of Calculus Try this

Assignment Lesson 5.4 Page 327 Exercises 1 – 49 odd