1. If a < b < c, then for any number b between a and c, the integral from a to c is the integral from a to b plus the integral from b to c. Theorem: Section 4.4 – Properties of Definite Integrals

2. Example:

3. Section 4.4 – Properties of Definite Integrals Example:

4. Section 4.4 – Properties of Definite Integrals Copyright  2010 Pearson Education, Inc. As the number of rectangles increased, the approximation of the area under the curve approaches a value.

5. Copyright  2010 Pearson Education, Inc. Section 4.4 – Properties of Definite Integrals

6. Example: Section 4.4 – Properties of Definite Integrals

7. Example: Section 4.4 – Properties of Definite Integrals Find the points of intersection

8. Average Value of a Continuous Function Copyright  2010 Pearson Education, Inc. Section 4.4 – Properties of Definite Integrals

9. Average Value of a Continuous Function Section 4.4 – Properties of Definite Integrals

10. a) Find the total profit from the first 10 days. b) Find the average daily profit from the first 10 days. Reminder: a)

11. Section 4.4 – Properties of Definite Integrals a) Find the total profit from the first 10 days. b) Find the average daily profit from the first 10 days. Reminder: b)

12. Section 4.4 – Properties of Definite Integrals

13. Differentiation Review: Copyright  2010 Pearson Education, Inc. Integration: Section 4.5 – Integration Techniques: Substitution

14. Copyright  2010 Pearson Education, Inc. Integration: Section 4.5 – Integration Techniques: Substitution

15. Copyright  2010 Pearson Education, Inc. Integrate: Section 4.5 – Integration Techniques: Substitution

16. Integrate: Section 4.5 – Integration Techniques: Substitution

17. Copyright  2010 Pearson Education, Inc. Integrate: Section 4.5 – Integration Techniques: Substitution

18. Integrate: Section 4.5 – Integration Techniques: Substitution

19. Copyright  2010 Pearson Education, Inc. Integrate: Section 4.5 – Integration Techniques: Substitution

20. Integrate: Section 4.5 – Integration Techniques: Substitution

21. Section 4.4 – Properties of Definite Integrals