3. 4-4: The Fundamental Theorems Definition: If f is continuous on [ a,b ] and F is an antiderivative of f on [ a,b ], then: The Fundamental Theorem:

4. Example Evaluate the definite integral using a Riemann Sum the fundamental theorem:

5. Try This Evaluate the definite integral using the fundamental theorem:

6. Try This Evaluate the definite integral using the fundamental theorem: 4

7. Try Me Evaluate the definite integral using the fundamental theorem: 1

8. Example Find the area bounded by,the x axis, and the lines x =-2 and x =2 -2 2

9. Try This Find the area bounded by, the x axis, the y axis and the line x =2. 0 2 sq. units

10. Important Idea Provided you can find an antiderivative of f, you can now evaluate a definite integral without using the limit of a Riemann sum.

11. Try This Evaluate: =0

12. Assignment 1. 291/1-21odd

13. Definition Write this down… If f is continuous on [ a,b ], then there exists a number c in [ a,b ] such that Mean Value Theorem for Integrals:

14. Analysis f( c) c a b The area of the rectangle is the same as the area under the curve from a to b The area of the rectangle is f (c)(b-a)

15. Example Find the value c guaranteed by the Mean Value Theorem for Integrals for f(x) over the specified interval: for [1,3]

16. Warm-Up Find the value c guaranteed by the Mean Value Theorem for Integrals for f(x) over the specified interval: for [1,3]

17. Definition If f is integrable over [ a,b ], then the average value of f on the interval [ a,b ] is:

18. Analysis Average Value a b f(x) Average Value Since area is height times width, area divided by width equals height which is average value.

19. Analysis area divided by width =average value

20. Example Find the average value of f (x)=4-x 2 on [0,3]

21. Try me Find the average value of f(x)=x 3 on [1,3]

22. Important Idea A definite integral is normally a number, but…sometimes it is helpful to write a definite integral as a function.

23. Example What is different? The answer is a function and not a number.

24. Example Find the area under and above the x axis and between the vertical lines x =0 & x =1; x =0 & x =2, and x =0 & x =3

25. Try This Evaluate: For

26. Definition If f is continuous on an open interval containing a, then for every x in the interval: The 2 nd Fund. Theorem: This is a constant

27. Example Integrate to find F as a function of x and then demonstrate the second fundamental theorem by differentiating the result:

28. Important Idea The second fundamental theorem states that the derivative of the integral of a function is the function evaluated at the upper limit of integration.

29. Definition If f is continuous on an open interval containing a, then for every x in the interval: The 2 nd Fund. Theorem: Chain rule version

30. Try This Evaluate:

31. Important Idea

32. Try This Evaluate:

34. Assignment 1. 291/1-21odd 2. 35-47 odd, 53-59 odd, 81-91 odd (slides 15-31)

35. Lesson Close Name and explain three important theorems mentioned in this lesson.