1. Aim: Riemann Sums & Definite Integrals Course: Calculus Do Now: Aim: What are Riemann Sums? Approximate the area under the curve y = 4 – x 2 for [-1, 1] using 4 inscribed rectangles.

2. Aim: Riemann Sums & Definite Integrals Course: Calculus Devising a Formula Using left endpoint to approximate area under the curve is lower sum a b 1 2 y n - 1 y n - 2 y0y0 y1y1 y n - 1 the more rectangles the better the approximation the exact area? take it to the limit! left endpoint formula

3. Aim: Riemann Sums & Definite Integrals Course: Calculus Using right endpoint to approximate area under the curve is Right Endpoint Formula upper sum a b y0y0 y1y1 y n - 1 ynyn right endpoint formula midpoint formula

4. Aim: Riemann Sums & Definite Integrals Course: Calculus Sigma Notation  sigma sum of terms The sum of the first n terms of a sequence is represented by where i is the index of summation, n is the upper limit of summation, and 1 is the lower limit of summation.

5. Aim: Riemann Sums & Definite Integrals Course: Calculus Summation Formulas

6. Aim: Riemann Sums & Definite Integrals Course: Calculus Riemann Sums A function f is defined on a closed interval [a, b]. It may have both positive and negative values on the interval. Does not need to be continuous. Δx1Δx1 Δx2Δx2 Δx3Δx3 Δx4Δx4 Δx5Δx5 Δx6Δx6 a = = b x0x0 x6x6 x1x1 x2x2 x3x3 x4x4 x5x5 Partition the interval into n subintervals not necessarily of equal length. a = x 0 < x 1 < x 2 <... < x n – 1 < x n = b - arbitrary/sample points for ith interval Δx i = x i – x i – 1

7. Aim: Riemann Sums & Definite Integrals Course: Calculus Riemann Sums Partition interval into n subintervals not necessarily of equal length. Δx1Δx1 Δx2Δx2 Δx3Δx3 Δx4Δx4 Δx5Δx5 Δx6Δx6 a = x0x0 x6x6 = b x1x1 x2x2 x3x3 x4x4 x5x5 - arbitrary/sample points for ith interval c i = x i

8. Aim: Riemann Sums & Definite Integrals Course: Calculus Riemann Sums Δx1Δx1 Δx2Δx2 Δx6Δx6 x6x6 = ba = x0x0 Δx4Δx4 Δx i = x i – x i – 1

9. Aim: Riemann Sums & Definite Integrals Course: Calculus Definition of Riemann Sum Let f be defined on the closed interval [a, b], and let Δ be a partition of [a, b] given by a = x 0 < x 1 < x 2 <.... < x n – 1 < x n = b, where Δx i is the length of the ith subinterval. If c i is any point in the ith subinterval, then the sum is called a Riemann sum for f for the partition Δ largest subinterval – norm - ||Δ|| or |P| equal subintervals – partition is regular regular partitiongeneral partition converse not true

10. Aim: Riemann Sums & Definite Integrals Course: Calculus Model Problem Evaluate the Riemann Sum R P for f(x) = (x + 1)(x – 2)(x – 4) = x 3 – 5x 2 + 2x + 8 on the interval [0, 5] using the Partition P with partition points 0 < 1.1 < 2 < 3.2 < 4 < 5 and corresponding sample points

11. Aim: Riemann Sums & Definite Integrals Course: Calculus Model Problem

12. Aim: Riemann Sums & Definite Integrals Course: Calculus Definition of Definite Integral If f is defined on the closed interval [a, b] and the limit exists, the f is integrable on [a, b] and the limit is denoted by The limit is called the definite integral of f from a to b. The number a is the lower limit of integration, and the number b is the upper limit of integration. Definite integral is a number Indefinite integral is a family of functions If a function f is continuous on the closed interval [a, b], then f is integrable on [a, b].

13. Aim: Riemann Sums & Definite Integrals Course: Calculus Evaluating a Definite Integral as a Limit c i = x i

14. Aim: Riemann Sums & Definite Integrals Course: Calculus Evaluating a Definite Integral as a Limit The Definite Integral as Area of Region If f is continuous and nonnegative on the closed interval [a, b], then the area of the region bounded by the graph of f, the x-axis and the vertical lines x = a and x = b is given by not the area

15. Aim: Riemann Sums & Definite Integrals Course: Calculus Properties of Definite Integrals

16. Aim: Riemann Sums & Definite Integrals Course: Calculus Areas of Common Geometric Figures Sketch & evaluate area region using geo. formulas. = 8 A = lw

17. Aim: Riemann Sums & Definite Integrals Course: Calculus Model Problems =0

18. Aim: Riemann Sums & Definite Integrals Course: Calculus Model Problem

19. Aim: Riemann Sums & Definite Integrals Course: Calculus Model Problem A1A1 A2A2 Total Area = -A 1 + A 2

20. Aim: Riemann Sums & Definite Integrals Course: Calculus Model Problem A1A1 A2A2

21. Aim: Riemann Sums & Definite Integrals Course: Calculus Model Problem take the limit n  

22. Aim: Riemann Sums & Definite Integrals Course: Calculus Definition of Riemann Sum

23. Aim: Riemann Sums & Definite Integrals Course: Calculus Definition of Riemann Sum

24. Aim: Riemann Sums & Definite Integrals Course: Calculus Subintervals of Unequal Lengths

25. Aim: Riemann Sums & Definite Integrals Course: Calculus Subintervals of Unequal Lengths