MCS 122 Chapter 5 Review of Basic Integration Some of the material in these slides is from Calculus 9/E by Howard Anton, Irl.
Download ReportTranscript MCS 122 Chapter 5 Review of Basic Integration Some of the material in these slides is from Calculus 9/E by Howard Anton, Irl.
MCS 122 Chapter 5 Review of Basic Integration Some of the material in these slides is from Calculus 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Antiderivatives Definition 5.2.1 (p. 322) Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Table 5.2.1 (p. 324) Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Theorem 5.2.3 (p. 325) Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Equations (4) – (7) (p. 326) Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Antiderivatives = Family of Functions 1 3 x dx 3 x C 2 Figure 5.2.1 (p. 327) Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Select the best answer for the following question. 7 2 1. Evaluate sec x dx 5x a) 7 x 2 tanx C 10 b) 7 ln x tanx C 5 c) 7 ln x tanx C 5 d) 7 ln x 1 sec3 x C 5 3 Question 1 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. The Area Problem 5.1.1 (p. 317) The Area Problem Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure 5.1.4 (p. 318) Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Rectangular Approximation for n Area = * f (x k 1 k ) x This is called a “Riemann Sum” for the area. Figure 5.4.4 (p. 344) Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Definition of Area for a continuous function Definition 5.4.3 (p. 344) Area Under a Curve Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure 5.1.4 (p. 318) Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Riemann sums yield “signed” area Figure 5.4.10 (p. 349) Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Three Common Approximations Figure 5.4.7 (p. 347) Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Approximations with variable width n f (x k 1 * k ) xk Figure 5.5.1 (p. 353) Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Definite Integral Definition 5.5.1 (p. 354) Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Continuous Functions are Integrable Theorem 5.5.2 (p. 355) Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Definite Integral Rules Definition 5.5.3 (p. 356) Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Definite Integral Rules Theorem 5.5.4 (p. 357) Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Definite Integral Rules Theorem 5.5.5 (p. 358) Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Definite Integral vs Antiderivatives We have seen two basic ideas so far: Antiderivative: Computes a family of functions f ( x)dx Definite Integral: Computes a number = area b a f ( x)dx Is there a connection between these two? Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Fundamental Theorem of Calculus - I Theorem 5.6.1 (p. 363) The Fundamental Theorem of Calculus, Part 1 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Fundamental Theorem of Calculus - I Practice Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Exercise 5.6.69 (p. 375) Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Fundamental Theorem of Calculus - II Theorem 5.6.3 (p. 370) The Fundamental Theorem of Calculus, Part 2 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. U-Substitution Guidelines for u-Substitution (p. 334) Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Select the best answer for the following question. 2. Evaluate 8 1 2 2 5x 3 4 x dx a) 89.5 b) 97.5 c) 96.5 Question 2 d) -89.5 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Select the best answer for the following question. 3. Find the area under the curve y a) 91 x over the interval [9, 16]. b) 24 2 3 c) 37 d) 24 2 3 Question 3 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Select the best answer for the following question. 4. Use part 2 of the Fundamental Theorem of Calculus to find: d x t3 dt 1 dx tant 1 2 3x a) sec2 x 1 3 t b) tant 1 2 3 2 c) 3x tanx 1 x sec x 1 tan2 x 1 d) 3 x tanx 1 Question 4 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Select the best answer for the following question. 6. Evaluate 2x 3 7 dx. a) 1 2x 3 8 C 8 b) 142x 36 C c) 1 2x 38 C 16 d) 1 2x 38 C 2 Question 6 Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Answers 1. c 2. a 3. b 4. d 5. 6. a Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.