1. The Definite Integral Objective: Introduce the concept of a “Definite Integral.”

2. Riemann Sums In our definition of net signed area, we assumed that for each positive number n, the interval [a, b] was subdivided into n subintervals of equal length. However, there may be functions where it is more convenient to use rectangles with different widths.

3. Riemann Sums In our definition of net signed area, we assumed that for each positive number n, the interval [a, b] was subdivided into n subintervals of equal length. However, there may be functions where it is more convenient to use rectangles with different widths. If we are to use rectangles of different widths, it it not enough to have n approach infinity. We need to avoid this situation.

4. Riemann Sums A partition of the interval [a, b] is a collection of points that divides [a, b] into n subintervals of lengths

5. Riemann Sums A partition of the interval [a, b] is a collection of points that divides [a, b] into n subintervals of lengths The partition is said to be regular provided the subintervals all have the same length

6. Riemann Sums For a regular partition, the widths of the approximating rectangles approach zero as n is made large. Since this need not be the case for a general partition, we need some way to measure the “size” of these widths.

7. Riemann Sums For a regular partition, the widths of the approximating rectangles approach zero as n is made large. Since this need not be the case for a general partition, we need some way to measure the “size” of these widths. One approach is to let denote the largest of the subinterval widths. The magnitude of is called the mesh size of the partition.

8. Definition 5.5.1 A function is said to be integrable on a finite closed interval [a, b] if the limit exists and does not depend on the choice of partitions or on the choice of the points in the subintervals.

9. Definition 5.5.1 When this is the case we denote the limit by the symbol which is called the definite integral of f from a to b. The numbers a and b are called the lower limit of integration and the upper limit of integration, respectively, and f(x) is called the integrand.

10. Riemann Sums This limit is called a Riemann Sum and the integral is sometimes called a Riemann integral in honor of the German mathematician Bernhard Riemann who formulated many of the basic concepts of calculus.

11. Theorem 5.5.2 If a function f is continuous on an interval [a, b], then f is integrable on [a, b], and the net signed area A between the graph of f and the interval [a, b] is

12. Example 1 Look at the following integrals and the graphs of the functions. The integral represents the area under the curve, between a and b.

13. Example 2 Evaluate

14. Example 2 Evaluate

15. Properties of the Definite Integral Definition 5.5.3 If a is in the domain of f, we define

16. Properties of the Definite Integral Definition 5.5.3 If a is in the domain of f, we define If f is integrable on [a, b], then we define

17. Theorem 5.5.4 If f and g are integrable on [a, b] and if c is a constant, then cf, f + g, and f – g are integrable on [a, b] and

18. Theorem 5.5.5 If f is integrable on a closed interval containing the three points a, b, and c, then no matter now the points are ordered.

19. Theorem 5.5.6 If f is integrable on [a, b] and f(x) > 0 for all x in [a, b], then

20. Theorem 5.5.6 If f is integrable on [a, b] and f(x) > 0 for all x in [a, b], then If f and g are integrable on [a, b] and f(x) > g(x) for all x in [a, b], then

21. Discontinuities and Integrability Definition 5.5.7 A function f that is defined on an interval I is said to be bounded on I if there is a positive number M such that for all x in the interval I. Geometrically, this means that the graph of f over the interval I lies between the lines y = -M and y = M.

22. Theorem 5.5.8 Let f be a function that is defined on the finite closed interval [a, b]. a)If f has finitely many discontinuities in [a, b] but is bounded on [a, b], then f is integrable on [a, b] b)If f is not bounded on [a, b], then f is not integrable on [a, b].

23. Homework Section 5.5 Pages 360-361 1, 5, 7, 9 17, 19, 21, 23