1. 5.3 Definite Integrals and Riemann Sums

2. I. Rules for Definite Integrals

5. II. Mean Value Theorem for Integrals A.) If f (x) is continuous on [a, b], then there exists at least one value c in [a, b] where

6. B.) Proof: Using Rule #7 Since f (x) is continuous on [a, b], f must take on every value between the minimum and maximum values of f. Therefore, there will be at least one value c in [a, b] where

7. C.) Graphically: a b c

8. D.) Average Velocity Using the Mean Value Theorem for Integrals:

9. III. Examples Find the following:

12. IV. Fundamental Theorem of Calculus Part II A.) If f(x) is continuous on [a, b] and F(x) is any antiderivative of f(x) on [a, b], then…

13. B.) Evaluate the following definite integrals using the Fundamental Theorem of Calculus:

14. C.) Ex. – Evaluate the following definite integrals using the Fundamental Theorem of Calculus

15. 1.)2.)3.)

16. D.) Find the average value of the following functions on the interval [1, 4].

17. E.) Ex.- Find the total distance traveled by a particle in rectilinear motion on [0,2] given the following velocity equation.

18. Total distance traveled is the area under the velocity curve bounded by the x-axis

19. F.) Ex.- Evaluate