1. 1. Does: ? 2. What is: ? Think about:

2. Finding Area between a Function & the x-axis Chapters 5.1 & 5.2 January 25, 2007

3. Using Data Underestimate the distance traveled: t: 0 2 4 6 8 10 v: 0 6.1 12.5 8.3 4.9 0

4. Ex: y=x 2 on [1,3] L 4 Length of subintervals (rectangles) = ∆x = (3 - 1)/4 =.5 Data points: x: 1 1.5 2 2.5 3 y: f(1) f(1.5) f(2) f(2.5) f(3) 1 9/4 4 25/4 9

5. L4L4

7. R4R4

10. T4T4

11. Trapezoid Rule: T n = (L n +R n )/2 OR T n = (∆x/2)[f(x 0 ) + 2f(x 1 ) + 2f(x 2 ) + … + 2f(x n-1 ) + f(x n )] Where n = # of sub-intervals (rectangles) ∆x = (b - a)/n

12. M4M4

13. Midpoint Rule: M n = ∆x[f(x 0 + ∆x/2) + f(x 1 + ∆x/2) + f(x 2 + ∆x/2) + … + f(x n-1 + ∆x/2) ] Where n = # of sub-intervals (rectangles) ∆x = (b - a)/n

14. You can use these approaches to find the area of “odd shapes”

15. L 10

16. R 10

17. T 10

18. Estimate the area using M 3 : F(x)=3/(1+x 2 ) on the interval [-1,5] Applet

19. Definition The area of A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles: where:

20. Use the Def n to find an expression for the exact area under: y=x 2 on the interval [1,3] f(x) = 3/(1+x 2 ) on the interval [-1, 5] g(x) = ln[x]/x on the interval [3,10]

21. Definition of Definite Integral: If f is a continuous function defined for a≤x≤b, we divide the interval [a,b] into n subintervals of equal width ∆x=(b- a)/n. We let x 0 (=a),x 1,x 2,…,x n (=b) be the endpoints of these subintervals and we let x 1 *, x 2 *, … x n * be any sample points in these subintervals so x i * lies in the ith subinterval [x i-1,x i ]. Then the Definite Integral of f from a to b is:

23. Use the Def n to find an expression for the exact area under: y=x 2 on the interval [1,3] f(x) = 3/(1+x 2 ) on the interval [-1, 5] g(x) = ln[x]/x on the interval [3,10]

24. Express the limit as a Definite Integral

25. Express the Definite Integral as a limit

26. Building on the idea of Area to evaluate the Definite Integral where c is a constant. Look at the graph…….

27. Building on the idea of Area to evaluate the Definite Integral Again, look at the graph…….

28. Building on the idea of Area to evaluate the Definite Integral Again, look at the graph…….

29. Building on the idea of Area to evaluate the Definite Integral Again, look at the graph…….

30. Building on the idea of Area to evaluate the Definite Integral Again, look at the graph…….

31. Properties of the Definite Integral