1. MTH 252 Integral Calculus Chapter 6 – Integration Section 6.6 – The Fundamental Theorem of Calculus Copyright © 2005 by Ron Wallace, all rights reserved.

2. Indefinite vs. Definite Integrals is the set of functions of the form F(x) + c where F’(x) = f(x) is the number How are these two entities related?

3. Reminder: The Mean-Value Theorem If f(x) is differentiable over (a,b), then there is a c  (a,b) where OR

4. Reminder: The Mean-Value Theorem If F(x) is differentiable over (x k-1,x k ), then there is a x k *  (x k-1,x k ) where OR

5. Reminder: Riemann Sum MVT: If F’(x) = f(x) … … Adding these up gives …

6. Reminder: Definite Integral If f(x) is continuous [a,b] and F’(x) = f(x) …

7. Fundamental Theorem of Calculus If f(x) is continuous [a,b] and F’(x) = f(x) … (Part I) Note that this works with ANY antiderivative. Why?

8. Integrating a Piecewise Function

9. Integrating an Absolute Value Function

10. Mean Value Theorem for Integrals Let f(x) be continuous over [a,b]. Extreme-Value Theorem   m & M where m ≤ f(x) ≤ M Therefore … Or …

11. Mean Value Theorem for Integrals Therefore … Intermediate-Value Theorem  f(x) takes on all values between m & M. Remember: m  f(x)  M Therefore, there exists x *  [a,b] where …

12. Mean Value Theorem for Integrals If f(x) is continuous over [a,b], then there exists x *  [a,b] where … OR The number f(x * ) is called the average value of the function. abx*x* f(x * )

13. The Area Function A(x)

14. The Derivative of the Area Function MVT for Integrals

15. Fundamental Theorem of Calculus (Part II) If f(t) is continuous over an interval containing a, then

16. FTC Part II: Examples Check these out by integrating and then differentiating! NOTE: Chain Rule!