# Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 1.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 1

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Chapter 5 The Definite Integral

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 5.1 Estimating with Finite Sums

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 4 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 5 Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 6 What you’ll learn about Distance Traveled Rectangular Approximation Method (RAM) Volume of a Sphere Cardiac Output … and why Learning about estimating with finite sums sets the foundation for understanding integral calculus.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 7 Example Finding Distance Traveled when Velocity Varies

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 8 Example Finding Distance Traveled when Velocity Varies

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 9 LRAM, MRAM, and RRAM approximations to the area under the graph of y=x 2 from x=0 to x=3

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 10 Example Estimating Area Under the Graph of a Nonnegative Function

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 5.2 Definite Integrals

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 12 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 13 Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 14 What you’ll learn about Riemann Sums The Definite Integral Computing Definite Integrals on a Calculator Integrability … and why The definite integral is the basis of integral calculus, just as the derivative is the basis of differential calculus.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 15 Sigma Notation

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 16 The Definite Integral as a Limit of Riemann Sums

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 17 The Existence of Definite Integrals

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 18 The Definite Integral of a Continuous Function on [a,b]

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 19 The Definite Integral

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 20 Example Using the Notation

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 21 Area Under a Curve (as a Definite Integral)

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 22 Area

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 23 The Integral of a Constant

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 24 Example Using NINT

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 5.3 Definite Integrals and Antiderivatives

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 26 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 27 Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 28 What you’ll learn about Properties of Definite Integrals Average Value of a Function Mean Value Theorem for Definite Integrals Connecting Differential and Integral Calculus … and why Working with the properties of definite integrals helps us to understand better the definite integral. Connecting derivatives and definite integrals sets the stage for the Fundamental Theorem of Calculus.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 29 Rules for Definite Integrals

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 30 Example Using the Rules for Definite Integrals

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 31 Example Using the Rules for Definite Integrals

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 32 Example Using the Rules for Definite Integrals

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 33 Average (Mean) Value

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 34 Example Applying the Definition

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 35 The Mean Value Theorem for Definite Integrals

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 36 The Mean Value Theorem for Definite Integrals

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 37 The Derivative of an Integral

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 38 Quick Quiz Sections 5.1 - 5.3

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 39 Quick Quiz Sections 5.1 - 5.3

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 40 Quick Quiz Sections 5.1 - 5.3

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 41 Quick Quiz Sections 5.1 - 5.3

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 42 Quick Quiz Sections 5.1 - 5.3

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 43 Quick Quiz Sections 5.1 - 5.3

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 5.4 Fundamental Theorem of Calculus

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 45 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 46 Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 47 What you’ll learn about Fundamental Theorem, Part 1 Graphing the Function Fundamental Theorem, Part 2 Area Connection Analyzing Antiderivatives Graphically … and why The Fundamental Theorem of Calculus is a Triumph of Mathematical Discovery and the key to solving many problems.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 48 The Fundamental Theorem of Calculus

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 49 The Fundamental Theorem of Calculus

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 50 Example Applying the Fundamental Theorem

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 51 Example The Fundamental Theorem with the Chain Rule

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 52 Example Variable Lower Limits of Integration

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 53 The Fundamental Theorem of Calculus, Part 2

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 54 The Fundamental Theorem of Calculus, Part 2

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 55 Example Evaluating an Integral

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 56 How to Find Total Area Analytically

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 57 How to Find Total Area Numerically

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 5.5 Trapezoidal Rule

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 59 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 60 Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 61 What you’ll learn about Trapezoidal Approximations Other Algorithms Error Analysis … and why Some definite integrals are best found by numerical approximations, and rectangles are not always the most efficient figures to use.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 62 Trapezoidal Approximations

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 63 The Trapezoidal Rule

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 64 Simpson’s Rule

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 65 Error Bounds

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 66 Quick Quiz Sections 5.4 and 5.5 x1467 f(x)10304020

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 67 Quick Quiz Sections 5.4 and 5.5 x1467 f(x)10304020

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 68 Quick Quiz Sections 5.4 and 5.5

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 69 Quick Quiz Sections 5.4 and 5.5

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 70 Quick Quiz Sections 5.4 and 5.5

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 71 Quick Quiz Sections 5.4 and 5.5

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 72 Chapter Test

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 73 Chapter Test

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 74 Chapter Test 8. A diesel generator runs continuously, consuming oil at a gradually increasing rate until it must be temporarily shut down to have the filters replaced. (a)Give an upper estimate and a lower estimate for the amount of oil consumed by the generator during that week. (b)Use the Trapezoidal Rule to estimate the amount of oil consumed by the generator during that week.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 75 Chapter Test

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 76 Chapter Test Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 77 Chapter Test Solutions 3 4 Let R be the region in the first quadrant enclosed by the x-axis and the graph of the function y  4x – x. 1. Sketch the rectangles and compute byhand the area for the MRAM approximations 4.125. 2. Sketch 4 the trapeziods and compute by hand the area for the T approximations. 3.75

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 78 Chapter Test Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 79 Chapter Test Solutions 8. A diesel generator runs continuously, consuming oil at a gradually increasing rate until it must be temporarily shut down to have the filters replaced. (a)Give an upper estimate and a lower estimate for the amount of oil consumed by the generator during that week. Upper = 4.392 L; Lower = 4.008 L (b)Use the Trapezoidal Rule to estimate the amount of oil consumed by the generator during that week. 4.2L

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 80 Chapter Test Solutions