New Symbol

1. Sec 5.2: The Definite Integral New Symbol Definition: the definite integral of ƒ over [a, b] Example: Find the definite integral of ƒ(x) = x + 2 over [ -1, 1 ] Solution:

2. Sec 5.2: The Definite Integral Note 1: integrand limits of integration lower limit a upper limit b Integral sign The dx simply indicates that the independent variable is x. The procedure of calculating an integral is called integration.

3. Sec 5.2: The Definite Integral definite integral has negative value the definite integral can be interpreted as the area under the curve A definite integral can be interpreted as a net area, that is,a difference of areas:

4. Sec 5.2: The Definite Integral Area under the curve Limit of the Riemann sum If you are asked to find one of them choose the easiest one. the definite integral of f from a to b three sides of the same coin

5. Sec 5.2: The Definite Integral Example: Evaluate the following integrals by interpreting each in terms of areas.

6. Sec 5.2: The Definite Integral Example: Evaluate the following integrals by interpreting each in terms of areas.

7. Sec 5.2: The Definite Integral Example: Evaluate the following integrals by interpreting each in terms of areas. Example: Evaluate the following integrals by interpreting each in terms of areas.

8. Sec 5.2: The Definite Integral Term-103

9. Sec 5.2: The Definite Integral Area under the curve the definite integral of f from a to b If you are asked to find one of them choose the easiest one.

10. Sec 5.2: The Definite Integral Express the limit as a definite integral on the given interval. the definite integral of f from a to b

11. Sec 5.2: The Definite Integral Definition: Definition: Riemann sum any sample points in these subintervals Remark: sample points lies in the i-th subinterval

12. Sec 5.2: The Definite Integral Definition: Riemann sum [a, b] divided into subintervals of equal width Example: Find the Riemann sum for ƒ(x) = x + 2 over [ 0, 5 ] Subintervals of unequal width divided into

13. Sec 5.2: The Definite Integral Area under the curve the definite integral of f from a to b If you are asked to find one of them choose the easiest one.

14. Sec 5.2: The Definite Integral Property (1) Example:

15. Sec 5.2: The Definite Integral Property (2)

16. Sec 5.2: The Definite Integral Property (3)

17. Sec 5.2: The Definite Integral Property (4) Property (5)

18. Sec 5.2: The Definite Integral

19. Sec 5.2: The Definite Integral Definition: Example: provided that this limit exists Find the definite integral of ƒ(x) = x + 2 over [ -1, 1 ] Definition: Solution: If the limit does exist, we say that the function f is integrable the limit exist, is integrable

20. Sec 5.2: The Definite Integral Theorem: If f (x) is continuous on [a, b] f (x) is integrable Example: is not integrable in [0, 1] Remark f(x) has only finite number of removable discontinuities Remark f(x) has only finite number of jump discontinuities

21. Sec 5.2: The Definite Integral

22. Sec 5.2: The Definite Integral Term-091

23. Sec 5.2: The Definite Integral Property

24. Sec 5.2: The Definite Integral Property (6)

25. Sec 5.2: The Definite Integral Property (7)

27. Sec 5.2: The Definite Integral

28. Sec 5.2: The Definite Integral Term-082

29. Sec 5.2: The Definite Integral