Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. 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. Definition (p. 322) Antiderivatives
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Table (p. 324)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Theorem (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. Figure (p. 327) Antiderivatives = Family of Functions
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. 1. Evaluate a) b) c) d) Question 1
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved (p. 317) The Area Problem 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 (p. 318)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure (p. 344) Rectangular Approximation for Area = This is called a “Riemann Sum” for the area.
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Definition (p. 344) Area Under a Curve Definition of Area for a continuous function
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure (p. 318)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure (p. 349) Riemann sums yield “signed” area
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure (p. 347) Three Common Approximations
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure (p. 353) Approximations with variable width
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Definition (p. 354) Definite Integral
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Theorem (p. 355) Continuous Functions are Integrable
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Definition (p. 356) Definite Integral Rules
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Theorem (p. 357) Definite Integral Rules
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Theorem (p. 358) Definite Integral Rules
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 Definite Integral: Computes a number = area 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. Theorem (p. 363) The Fundamental Theorem of Calculus, Part 1 Fundamental Theorem of Calculus - I
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 (p. 375)
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Theorem (p. 370) The Fundamental Theorem of Calculus, Part 2 Fundamental Theorem of Calculus - II
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Guidelines for u-Substitution (p. 334) U-Substitution
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 a) 89.5 b) 97.5 c) 96.5 d) Question 2
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 over the interval [9, 16]. a) 91 b) c) 37 d) 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: a) b) c) d) 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 a) b) c) d) Question 6
Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. 1.c 2.a 3.b 4.d 5. 6.a Answers