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Drill: Evaluate each sum

Drill: Evaluate each sum. 1 2 + 2 2 + 3 2 + 4 2 + 5 2 =55 [3(0) -2] + [3(1) -2] + [3(2) -2] + [3(3) -2] + [3(4) -2] = 20 100 (0 + 1) 2 + 100 (1 + 1) 2 + 100 (2 + 1) 2 + 100(3 + 1) 2 + 100 (4 + 1) 2 =5500. Write the sum in sigma notation. 1 + 2 + 3 + …..+ 98 + 99

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Drill: Evaluate each sum

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  1. Drill: Evaluate each sum • 12 + 22 + 32 + 42 + 52 =55 • [3(0) -2] + [3(1) -2] + [3(2) -2] + [3(3) -2] + [3(4) -2] = 20 • 100 (0 + 1)2 + 100 (1 + 1)2 + 100 (2 + 1)2 + 100(3 + 1)2 + 100 (4 + 1)2 =5500

  2. Write the sum in sigma notation • 1 + 2 + 3 + …..+ 98 + 99 • 0 + 2 + 4 + ….48 + 50 • 3(1)2 + 3(2)2 + …. 3(500)2

  3. Definite Integrals Lesson D.3

  4. Objectives • Students will be able to • express the area under a curve as a definite integral and as a limit of Riemann sums. • compute the area under a curve using a numerical integration procedure.

  5. Key Concept: Riemann Sum A Riemann sum, Rn, for function f on the interval [a, b] is a sum of the form where the interval [a, b] is partitioned into n subintervals of widths Δxk, and the numbers {ck} are sample points, one in each subinterval.

  6. Example: Calculating Riemann Sums Upper = using right endpoints: ¼ (1/64+ 1/8 + 27/64 + 1) = 25/64 Lower = using left endpoints: ¼ ( 1/64 + 1/8 + 27/64) = 9/64

  7. More Rectangles More accuracy

  8. Definite Integral The function is the integrand Upper limit of integration Let f be continuous on [a,b] and be partitioned into n subintervals of equal length Δx = (b – a)/n ck is some point in the kth subinterval. When you find the value of the integral, you have evaluated the integral Integral sign x is the variable of integration lower limit of integration

  9. Express each limit as a definite integral

  10. Key Concept: Area Under a Curve If y = f (x) is nonnegative and integrable on [a, b], and if Rn is any Riemann sum for f on [a, b], then

  11. Use the graph of the integrand and areas to evaluate the integral + 3) dx 5 2 6 A=1/2(6)(2+5)= 21

  12. Example Area Under a Curve Write the definite integral and determine the area under the curve over the interval [–4, 4]. 4 -4

  13. Homework D.3 Day 1 • Pg. 278 • # 9-18, 23, 25, 27, 31

  14. Daily • What does the following mean:

  15. Area under a curve (as a definite integral) • If y = f(x) is a nonnegative and integrable over a closed [a,b], then the area under the curve y = f(x) from a to b is the intergral of f from a to b. • A =

  16. Area below x axis

  17. Integration Properties • - when f(x) ≤ 0 (below x axis) • = (area above the x axis) – (area below the x axis) • If f is defined at x = a, then

  18. Integration Properties Cont. • If f is integrable on the three closed intervals determined by a, b, and c then

  19. Integration Properties Examples • = -

  20. Integration Properties Examples • If f is defined at x = a, then • = 0

  21. Integration Properties Examples • If f is integrable on the three closed intervals determined by a, b, and c then • dx =

  22. Integration Properties Examples • Suppose • Find:

  23. Homework D.3 Day 2 • Pg. 279 • #33-35, 41 - 44

  24. Daily • Evaluate the integral using what you know…

  25. Area below x axis

  26. Evaluate each definite integral using geometric formulas.

  27. Evaluating Integrals • In order to evaluate a definite integral you will find the antiderivative and then evaluate at b then subtract the evaluation at a. • = F(b) – F(a)

  28. Example • dx Antiderivative of is x³ So dx = = 4³ - 2³ = 64 – 8 = 56

  29. Example • Antiderivative of 3 is 3x • = 3(10) – 3(4) • 30 – 12 = 18

  30. Homework D.3 Day 3 • Pg. 278-280 • #3-8, 23, 25, 27 (compare to Day 1 answers once evaluated), 47 a-d, 48 a-f, 51, 52

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