Presentation on theme: "The Fundamental Theorem of Calculus Lesson 7.4. 2 Definite Integral Recall that the definite integral was defined as But … finding the limit is not often."— Presentation transcript:
2 Definite Integral Recall that the definite integral was defined as But … finding the limit is not often convenient We need a better way!
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 !
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
5 You Gotta Try It Consider What is F(x), the antiderivative? Evaluate F(5) – F(0)
6 Properties Bringing out a constant factor The integral of a sum is the sum of the integrals
7 Properties Splitting an integral f must be continuous on interval containing a, b, and c
8 Example Consider Which property is being used to find F(x), the antiderivative? Evaluate F(4) – F(0)
9 Try Another What is Hint … combine to get a single power of x What is F(x)? What is F(3) – F(1)?
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
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