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 transcript:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd EXAMPLE 37-Page 875 Approximately 12 squares y = x 2 /

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

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

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

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

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

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

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

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

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.