1. 5.2 Riemann Sums and Area

2. I. Riemann Sums A.) Let f (x) be defined on [a, b]. Partition [a, b] by choosing These partition [a, b] into n parts of length Δx 1, Δx 2,… Δx n. In each subinterval, choose a point c 1, c 2,…c n and form the sum This is called a Riemann Sum *NOTE: LRAM, MRAM, and RRAM are all Riemann sums.

3. B.) Def.- The NORM of P, denoted ||P||, is the length of the largest subinterval. C.) Def. – If is defined on [a, b], then the DEFINITE INTEGRAL of f (x) over [a, b] is provided the limit exists. This is denoted *NOTE: If the limit exists, then f (x) is said to be integrable on [a, b].

4. II. EXISTENCE A.) Thm: (Existence of the limit of a Riemann Sum) 1.) If f (x) is continuous on [a, b], then f (x) is integrable on [a, b]. 2.) If f (x) is bounded and has a finite number of discontinuities on [a, b], then f (x) is integrable on [a, b].

5. III. Examples A.) Express the limit as a definite integral:, where P is a partition of [1, 4]., where P is a partition of [1, 4].

6. B.) Express as a limit. where P is a partition of.

7. III. Definite Integrals and Technology A.) fnInt – Under MATH – 9 where f (x) is the function, x is the variable of integration, and a and b are the bounds of the integration.

8. B.) Use your calculator to evaluate the following definite integrals:

10. V. Area Under a Curve A.) Def.- If f (x) ≥ 0 and integrable on [a, b], then the area bounded by the curve, the x-axis, and the vertical lines x = a and x = b is

11. B.) Find the area under the curve from x = 0 to x = 3.

12. C.) Find the area bounded by the curve the x-axis, and the vertical lines x = -1 to x = 3. By Geometry - By Definite Integral - WHY????-

13. D.) FACT – For any integrable function: **Note – AREA is ALWAYS POSITIVE, the definite integral MAY be NEGATIVE.

14. VI. Examples A.) Determine the value of the following definite integrals by using the areas bounded by the graph of the function, the x-axis, and the bounds given.

15. WHY???

16. WHY???

17. WHY???

18. B.) Express the area bounded by f (x) and the x- axis in terms of an integrable expression and then find it.