The First Fundamental Theorem of Calculus

1. The First Fundamental Theorem of Calculus

2. First FTOC ** Notice!! You do not have to include a “C” when you integrate f(x)

3. Example Evaluating an Integral

5. Example

6. The Derivative of an Integral

7. Now let’s see some more examples / practice of definite integral problems

8. The Mean Value Theorem for Definite Integrals

9. The Mean Value Theorem for Definite Integrals

10. Mean Value Theorem (for definite integrals) If f is continuous on then at some point c in , The mean value theorem for definite integrals says that for a continuous function, at some point on the interval the actual value will equal the average value. p

11. Average(Mean) Value

12. Example Applying the Definition

13. The average value of a function is the value that would give the same area if the function was a constant:

14. Now let’s see some examples using the Mean Value Theorem for Integrals and to find the average value of a function

15. The Second Fundamental Theorem of Calculus

16. The Second Fundamental Theorem of Calculus

17. The Derivative of an Integral

18. 1. Derivative of an integral. Second Fundamental Theorem:

19. Second Fundamental Theorem: 1. Derivative of an integral. 2. Derivative matches upper limit of integration.

20. Scond Fundamental Theorem: 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.

21. Second Fundamental Theorem: New variable. 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.

22. Example

23. Example Applying the Fundamental Theorem