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. Antiderivatives Definition: A function F is called an antiderivative of a function f on an interval I if F’(x) = f(x) for all x in I. Definition: The notation is used for an antiderivative of f and is called an indefinite integral.

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

Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. 1. Evaluate a) b) c) d) Question 1 Integration Practice – Solutions at end of Slides

Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. The Area Problem

Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. To compute area we use the idea of approximating rectangles.

Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. 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 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. Riemann sums yield “signed” area Riemann Sum area = area above x-axis – area below x-axis

Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Three Common Approximations

Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Definite Integral Definition of Definite Integral: If f(x) is a function defined for a ≤ x ≤ b, we divide the inetrval [a,b] into n subintervals of equal width Δx = (b-a)/n. We let x 0 =a, x 1, x 2, …, x n (=b) be the endpoints of these subintervals and we let x 1 *, x 2 *, …, x n * be any sample points in these subintervals. Then, the definite integral of f from a to b is provided this limit exists and gives the same value for all possible choices of sample points. If the limit exists, we say f is integrable on [a,b]

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

Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. 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 Rules

Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. 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. Fundamental Theorem of Calculus – Part I

Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. (Also called the Net Change Theorem in Stewart) Fundamental Theorem of Calculus – Part II

Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. 2. Evaluate a) 89.5 b) 97.5 c) 96.5 d) Question 2 Integration Practice – Solutions at end of Slides

Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. 3. Find the area under the curve over the interval [9, 16]. a) 91 b) c) 37 d) Question 3 Integration Practice – Solutions at end of Slides

Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. 4. Use part 2 of the Fundamental Theorem of Calculus to find: a) b) c) d) Question 4 Integration Practice – Solutions at end of Slides

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 Answers