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Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. Integration Prepared by: Midori Kobayashi Humber.

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Presentation on theme: "Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. Integration Prepared by: Midori Kobayashi Humber."— Presentation transcript:

1 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. Integration Prepared by: Midori Kobayashi Humber College C30

2 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.1 30.1 The Indefinite Integral

3 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.1-EXAMPLE 2-Page 857 x 3 + C derivative 3x23x2 antiderivative

4 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.1-EXAMPLE 6-Page 859

5 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.1-EXAMPLE 7-Page 859

6 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.1-EXAMPLE 10-Page 860

7 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.1-EXAMPLE 13-Page 861 1  4/3

8 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.1-EXAMPLE 14-Page 861

9 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.1-EXAMPLE 17-Page 862

10 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.2 30.2 Rules for Finding Integrals

11 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.2-EXAMPLE 19-Page 864 u du n=6

12 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.2-EXAMPLE 19-Page 864-Continued u=x 3 +3

13 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.2-EXAMPLE 20-Page 865 Checked!

14 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.2-EXAMPLE 21-Page 865 u ≠ du = 3x 2 dx Need to multiply by 3 3 Need to divide by 3 ⅓ du

15 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.2-EXAMPLE 21-Page 865-Continued n=6 Bring u back! u = x 3 +3 u du

16 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.2-EXAMPLE 24-Page 867 u ≠ du = -2xdx Need to multiply by – 2 – 2 Need to divide by – 2 –½–½ du

17 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.2-EXAMPLE 24-Page 867-Continued Bring u back! u = 3 – x 2 u du

18 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.3 30.3 Constant of Integration

19 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.3-EXAMPLE 29-Page 870 Parabola opening upward y = x 2 + C

20 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.3-EXAMPLE 30-Page 870 y = x 2 + C (2,9)● C=5

21 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.3-EXAMPLE 32-Page 872 y’ =  y”dx and y =  y’ dx Substituting

22 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.3-EXAMPLE 32-Page 872-Continued y’ =  y”dx and y =  y’ dx

23 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.4 30.4 The Definite Integral

24 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.4-EXAMPLE 33-Page 874 upper limit Antiderivative of x 2 lower limit

25 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.4-EXAMPLE 34-Page 874 Antiderivative of u 3 u du 2 ½

26 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.4-EXAMPLE 35-Page 874 Antiderivative of 1/x Absolute value

27 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.4-EXAMPLE 36-Page 875 The function 1/x is discontinuous at x = 0

28 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.5 30.5 Approximate Area Under a Curve

29 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.5-EXAMPLE 37-Page 875 Approximately 12 squares y = x 2 /3 12 5 34 67 89 10 11 12

30 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.5-EXAMPLE 38-Page 876 R

31 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.5-EXAMPLE 39-Page 876 Summation = Adding up from n=1 to n=4

32 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.5-EXAMPLE 41-Page 878 Midpoints 13579

33 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.5-EXAMPLE 41-Page 878 rectangle 13579

34 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.6 30.6 Exact Area Under a Curve

35 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.6-EXAMPLE 43-Page 882

36 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.6-EXAMPLE +

37 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 30.6-EXAMPLE ++

38 Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. Copyright Copyright © 2008 John Wiley & Sons Canada, Ltd. All rights reserved. Reproduction or translation of this work beyond that permitted by Access Copyright (The Canadian Copyright Licensing Agency) is unlawful. Requests for further information should be addressed to the Permissions Department, John Wiley & Sons Canada, Ltd. The purchaser may make back-up copies for his or her own use only and not for distribution or resale. The author and the publisher assume no responsibility for errors, omissions, or damages caused by the use of these programs or from the use of the information contained herein.

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