Chapter 5-The Integral Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

Chapter 5-The Integral 5.1 Introduction to Integration- The Area Problem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Summation Notation EXAMPLE: Evaluate the sums:

Chapter 5-The Integral 5.1 Introduction to Integration-The Area Problem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Some Special Sums EXAMPLE: Calculate the following sums:

Chapter 5-The Integral 5.1 Introduction to Integration-The Area Problem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Some Special Sums EXAMPLE: Suppose that x 0, x 1, …, x N are points in the domain of a function F. Show that

Chapter 5-The Integral 5.1 Introduction to Integration-The Area Problem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Approximation of Area DEFINITION: Let N be a positive integer. The uniform partition P of order N of the interval [a, b] is the set {x 0, x 1,..., x N } of N + 1 equally spaced points such that

Chapter 5-The Integral 5.1 Introduction to Integration-The Area Problem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Approximation of Area EXAMPLE: Using the uniform partition of order 6 of the interval [1, 4], find the right endpoint approximation to the area A of the region bounded above by the graph of f(x) = x 2 + x, below by the x- axis, on the left by the vertical line x = 1, and on the right by the vertical line x = 4.

Chapter 5-The Integral 5.1 Introduction to Integration-The Area Problem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved A Precise Definition of Area DEFINITION: The area A of the region that is bounded above by the graph of f, below by the x-axis, and laterally by the vertical lines x = a and x = b, is defined as the limit

Chapter 5-The Integral 5.1 Introduction to Integration-The Area Problem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved A Precise Definition of Area EXAMPLE: Calculate the area A of the region that is bounded above by the graph of y = 2x, below by the x-axis, and on the sides by the vertical lines x = 0 and x = 6.

Chapter 5-The Integral 5.1 Introduction to Integration-The Area Problem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. Calculate 2. Calculate 3. What is the fifth subinterval of [3, 9] when the uniform partition of order 7 is used? 4. What is the right endpoint approximation of the area of the region under the graph of y = 1/x and over the interval [1, 2] when the uniform partition of order 3 is used?

Chapter 5-The Integral 5.2 The Riemann Integral Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Riemann Sums EXAMPLE: Write a Riemann sum for the function f(x) = x 2 − 4 and the interval [a, b] = [−5, 3] using the partition {−5,−3,−1, 1, 3}.

Chapter 5-The Integral 5.2 The Riemann Integral Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Riemann Sums EXAMPLE: Find the upper and lower Riemann sums of order 4 for f(x) = x 2 − 4 over the interval [-5, 3]

Chapter 5-The Integral 5.2 The Riemann Integral Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Riemann Sums THEOREM: Suppose that f is continuous on an interval [a, b]. a) If S N = {s 1,..., s N } is any choice of points associated with the uniform partition of order N, then b) The numbers R(f,L N ) and R(f, U N ) become arbitrarily close to each other for N sufficiently large. That is,

Chapter 5-The Integral 5.2 The Riemann Integral Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Riemann Integral DEFINTION: Suppose that f is a function defined on the interval [a, b]. We say that the Riemann sums R(f, S N ) tend to the real number as N tends to infinity, if, for any  > 0, there is a positive integer M such that for all N greater than or equal to M and any choice of S N. If this is the case, we say that f is integrable on [a, b], and we denote the limit by the symbol

Chapter 5-The Integral 5.2 The Riemann Integral Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Riemann Integral THEOREM: If f is continuous on the interval [a, b], then f is integrable on [a, b]. That is, the Riemann Integral exists.

Chapter 5-The Integral 5.2 The Riemann Integral Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Calculating Riemann Integrals THEOREM: Suppose that F is an antiderivative of a continuous function f on [a, b]. Then, EXAMPLE: Evaluate

Chapter 5-The Integral 5.2 The Riemann Integral Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Using the Fundamental Theorem of Calculus to Compute Areas EXAMPLE: Calculate the area A of the region that lies under the graph of f(x) = x 2, above the x-axis, and between the vertical lines x = 2 and x = 6.

Chapter 5-The Integral 5.2 The Riemann Integral Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. If {s 1, s 2, s 3, s 4 } is a choice of points associated with the order 4 uniform partition of the interval [−1, 2], what is the smallest possible value that can be chosen for s 3 ? The largest? 2. What are the lower and upper Riemann sums for f(x) = x 2 when the order 4 uniform partition of [−1, 2] is used? 3. Evaluate 4. True or false: 5. True or false:

Chapter 5-The Integral 5.3 Rules for Integration Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved THEOREM: If f and g are integrable functions on the interval [a,b] and if  is constant, then f + g, f-g, and  f are integrable and

Chapter 5-The Integral 5.3 Rules for Integration Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Calculate the integral EXAMPLE: Suppose that g is a continuous function on the interval [4, 9] that satisfies Calculate

Chapter 5-The Integral 5.3 Rules for Integration Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Reversing the Direction of Integration EXAMPLE: Evaluate

Chapter 5-The Integral 5.3 Rules for Integration Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Reversing the Direction of Integration THEOREM: If f is continuous on an interval that contains the points a and b (in any order) and if F is an antiderivative of f, then THEOREM: If f is continuous on an interval that contains the three points a, b, and c (in any order), then

Chapter 5-The Integral 5.3 Rules for Integration Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Order Properties of Integrals THEOREM: If f, g, and h are integrable on [a, b] and g(x) ≤ f(x) ≤ h(x) for x in [a, b], then In particular, if f is integrable on [a, b] and if m and M are constants such that m ≤ f(x) ≤ M, for all x in [a, b], then

Chapter 5-The Integral 5.3 Rules for Integration Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Order Properties of Integrals EXAMPLE: Estimate THEOREM: If f is integrable on [a,b] and if f(x) ≥ 0, then THEOREM: If f is integrable on [a,b], then

Chapter 5-The Integral 5.3 Rules for Integration Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Mean Value Theorem for Integrals THEOREM: Let f be continuous on [a,b]. There is a value c in (a,b) such that

Chapter 5-The Integral 5.3 Rules for Integration Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. What is 2. Calculate 3. If and then what is 4. What is the average value of f(t) = t 2 for 1 ≤t ≤ 4?

Chapter 5-The Integral 5.4 The Fundamental Theorem of Calculus Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved THEOREM: Let f be continuous on [a,b]. a) If F is any antiderivative of f on [a,b], then b) The function F defined by is an antiderivative of f: F is continuous on [a,b] and

Chapter 5-The Integral 5.4 The Fundamental Theorem of Calculus Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Examples Illustrating the First Part of the Fundamental Theorem EXAMPLE: Compute EXAMPLE: Calculate

Chapter 5-The Integral 5.4 The Fundamental Theorem of Calculus Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Examples Illustrating the Second Part of the Fundamental Theorem EXAMPLE: The hydrostatic pressure exerted against the side of a certain swimming pool at depth x is What is the rate of change of P with respect to depth when the depth is 4?

Chapter 5-The Integral 5.4 The Fundamental Theorem of Calculus Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Examples Illustrating the Second Part of the Fundamental Theorem EXAMPLE: Calculate the derivative of with respect to x.

Chapter 5-The Integral 5.4 The Fundamental Theorem of Calculus Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. Suppose that What is F’(2)? 2. Suppose that What is F’(2)? 3. Suppose that What is F’(1)? 4. Evaluate

Chapter 5-The Integral 5.5 A Calculus Approach to the Logarithm and Exponential Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

Chapter 5-The Integral 5.5 A Calculus Approach to the Logarithm and Exponential Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved THEOREM: The natural logarithm has the following properties:

Chapter 5-The Integral 5.5 A Calculus Approach to the Logarithm and Exponential Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Properties of the Natural Logarithm THEOREM: Let x and y be positive and p be any number. Then

Chapter 5-The Integral 5.5 A Calculus Approach to the Logarithm and Exponential Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Graphing the Natural Logarithm THEOREM: The natural logarithm is an increasing function with domain equal to the set of positive real numbers. Its range is the set of all real numbers. The equation ln (x) = has a unique solution x element of the positive reals for every real number. The graph of the natural logarithm function is concave down. The y-axis is a vertical asymptote.

Chapter 5-The Integral 5.5 A Calculus Approach to the Logarithm and Exponential Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Exponential Function We define b = exp (a) if and only if a=ln(b)

Chapter 5-The Integral 5.5 A Calculus Approach to the Logarithm and Exponential Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Properties of the Exponential Function exp(s+t) = exp(s)  exp(t) exp(s-t) = exp(s)/exp(t) (exp(s)) t = exp(st)

Chapter 5-The Integral 5.5 A Calculus Approach to the Logarithm and Exponential Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Derivatives and Integrals Involving the Exponential Function THEOREM: The exponential function satisfies

Chapter 5-The Integral 5.5 A Calculus Approach to the Logarithm and Exponential Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Number e Define e = exp(1) so ln( e ) = 1

Chapter 5-The Integral 5.5 A Calculus Approach to the Logarithm and Exponential Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Logarithms and Powers with Arbitrary Bases a x = exp(x  ln(a)) EXAMPLE: Use the above formula to derive the Power Rule of differentiation

Chapter 5-The Integral 5.5 A Calculus Approach to the Logarithm and Exponential Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Logarithms and Powers with Arbitrary Bases THEOREM: If a,b>0 and x,y are real numbers, then

Chapter 5-The Integral 5.5 A Calculus Approach to the Logarithm and Exponential Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Logarithms with Arbitrary Bases THEOREM: For any fixed positive a ≠ 1, the function x  a x has domain of the reals and image of positive reals. The function x  log a (x) has domain positive reals and image reals. The two functions are inverses of each other.

Chapter 5-The Integral 5.5 A Calculus Approach to the Logarithm and Exponential Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Logarithms with Arbitrary Bases THEOREM: If x, y, a>0 and a≠ 1, then

Chapter 5-The Integral 5.5 A Calculus Approach to the Logarithm and Exponential Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Differentiation and Integration of a x and log a (x)

Chapter 5-The Integral 5.5 A Calculus Approach to the Logarithm and Exponential Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. Suppose that and What is 2. For what value of a is 3. Suppose that a is a positive constant. For what value of u is a x = e u ? 4. Suppose that a is a positive constant not equal to 1. For what value of k is log a (x) = k · ln(x)?

Chapter 5-The Integral 5.6 Integration by Substitution Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Key Steps for the method of substitution 1.Find a expression  (x) in the integrand that has the derivative  ’(x) that also appears in the integrand. 2.Substitute u for  (x) and du for  ’(x) dx 3.Do not proceed unless the entire integrand is expressed in terms of the new variable u. 4.Evaluate the new integral to obtain an answer expressed in terms of u. 5.Resubstitute to obtain an answer in terms of x.

Chapter 5-The Integral 5.6 Integration by Substitution Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Some Examples of Indefinite Integration by Substitution EXAMPLE: Evaluate the indefinite integral  sin 4 (x) cos(x) dx. EXAMPLE: Compute the integral

Chapter 5-The Integral 5.6 Integration by Substitution Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Method of Substitution for Definite Integrals EXAMPLE: Evaluate the integral EXAMPLE: Calculate

Chapter 5-The Integral 5.6 Integration by Substitution Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved When an Integration Problem Seems to Have Two Solutions EXAMPLE: Compute the indefinite integral  sin(x)cos(x) dx.

Chapter 5-The Integral 5.6 Integration by Substitution Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Integral Tables EXAMPLE: Use integral tables to compute

Chapter 5-The Integral 5.6 Integration by Substitution Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Integrating Trigonometric Functions EXAMPLE: Show that  tan(x) dx = ln(|sec(x)|) + C EXAMPLE: Use a substitution to derive the formula  sec(x) dx = ln(|sec(x)+tan(x)|) + C

Chapter 5-The Integral 5.6 Integration by Substitution Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. The method of substitution is the antiderivative form of what differentiation rule? 2. Evaluate  cos3(x) sin(x) dx. 3. Evaluate 4. Evaluate

Chapter 5-The Integral 5.7 More on the Calculation of Area Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Rules for Calculating Area (i) If f(x) ≥0 for x in [a, b], then equals the area of the region that is under the graph of f, above the x-axis, and between x = a and x = b. (ii) If f(x) ≤ 0 for x in [a, b], then equals the negative of the area of the region that is above the graph of f, below the x-axis, and between x = a and x = b.

Chapter 5-The Integral 5.7 More on the Calculation of Area Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: What is the area of the region bounded by the graph f(x) = sin(x) and the x-axis between the limits x=  /3 and 3  /2?

Chapter 5-The Integral 5.7 More on the Calculation of Area Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved THEOREM: Let f and g be continuous functions on the interval [a,b] and suppose that f(x)≥g(x) for all x in [a,b]. The area of the region under the graph of f and above the graph of g on the interval [a,b] is given by The Area between Two Curves

Chapter 5-The Integral 5.7 More on the Calculation of Area Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Calculate the area A of the region between the curves f(x) = −x and g(x) = 3x 2 − 8 for x in the interval [−1, 1]. The Area between Two Curves

Chapter 5-The Integral 5.7 More on the Calculation of Area Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Compute the area between the curves x = y 2 − y − 4 and x = −y 2 + 3y Reversing the Roles of the Axes

Chapter 5-The Integral 5.7 More on the Calculation of Area Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 1. Suppose that f and g are continuous functions on [a, b] with g(x) ≤ f(x) for every x in [a, b]. The area of the region that is bounded above and below by the graphs of f and g respectively is equal for what expression h(x)? 2. What is the area between the x-axis and the curve y = sin(x) for 0 ≤ x ≤ 2  ? 3. What is the area between the y-axis and the curve x = 1−y 2 ? 4. The area of a region is obtained as by integrating with respect to x. What integral represents the area if the integration is performed with respect to y Quick Quiz

Chapter 5-The Integral 5.8 Numerical Techniques of Integration Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Midpoint Rule Let f be a continuous function on the interval [a, b]. Le N be a positive integer. For each integer j with 1≤ j ≤N, let and let M N be the midpoint approximation If C is a constant such that |f’’(x)| ≤C for a ≤x ≤b, then

Chapter 5-The Integral 5.8 Numerical Techniques of Integration Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Midpoint Rule EXAMPLE: Let f(x) = 1/(1+x 2 ). Estimate using the Midpoint Rule with N=4. Based on this approximation, how small might the exact value of A be? How large?

Chapter 5-The Integral 5.8 Numerical Techniques of Integration Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Trapezoidal Rule THEOREM: Let f be a continuous function on the interval [a, b]. Let N be a positive integer. Let be the trapezoidal approximation. If |f’’(x)| ≤ C for all x in the interval [a,b], then the order N trapezoidal approximation is accurate to within C(b-a) 3 /(12N 2 ).

Chapter 5-The Integral 5.8 Numerical Techniques of Integration Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Simpson’s Rule THEOREM: Let f be continuous on the interval [a, b]. Let N be a positive even integer. If C is any number such that |f (4) (x)| ≤ C for a ≤ x ≤ b, then where

Chapter 5-The Integral 5.8 Numerical Techniques of Integration Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Simpson’s Rule EXAMPLE: Let f(x) = 1/(1 + x 2 ). Estimate using Simpson’s Rule with N =4. In what interval of real numbers does A lie?

Chapter 5-The Integral 5.8 Numerical Techniques of Integration Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. Approximate by the Midpoint Rule with N = Approximate by the Trapezoidal Rule with N = Approximate by Simpson’s Rule with N = Approximate by a) the Midpoint Rule, b) the Trapezoidal Rule, and c) Simpson’s Rule, using, in each case, as much of the data, f(2.5) = 24, f(3.0) = 36, and f(3.5) = 12, as possible.