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

Chapter 5-The Integral

Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

Chapter 5-The Integral5.1 Introduction to Integration-The Area Problem


Summation Notation

EXAMPLE: Evaluate the sums:



Some Special Sums

EXAMPLE: Calculate the following sums:



Some Special Sums

EXAMPLE: Suppose that x0, x1, …, xN are points in the domain of a function F. Show that



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{x0, x1, . . . , xN} of N + 1 equally spaced points such that



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) = x2 + 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.



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



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 thex-axis, and on the sides by the vertical lines x = 0 and x = 6.



Quick Quiz

1. Calculate 2. Calculate3. 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 overthe interval [1, 2] when the uniform partition of order 3 is used?

Chapter 5-The Integral5.2 The Riemann Integral


Riemann Sums

EXAMPLE: Write a Riemann sum for the function f(x) = x2 − 4 and the interval [a, b] = [−5, 3] using the partition {−5,−3,−1, 1, 3}.



Riemann Sums

EXAMPLE: Find the upper and lower Riemann sums of order 4 for f(x) = x2 − 4 over the interval [-5, 3]



Riemann Sums

THEOREM: Suppose that f is continuous on an interval [a, b].a) If SN = {s1, . . . , sN} is any choice of points associated with the uniform partition of order N, then

b) The numbers R(f,LN) and R(f, UN) become arbitrarily close to each other for N sufficiently large. Thatis,



The Riemann IntegralDEFINTION: Suppose that f is a function defined on the interval [a, b]. We say that the Riemann sumsR(f, SN) tend to the real number as N tends toinfinity, 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 SN. If this is the case, we say that f is integrable on[a, b], and we denote the limit by the symbol



The Riemann Integral

THEOREM: If f is continuous on the interval [a, b], then f is integrable on [a, b]. That is, the RiemannIntegral exists.



Calculating Riemann IntegralsTHEOREM: Suppose that F is an antiderivative of a continuous function f on [a, b]. Then,

EXAMPLE: Evaluate



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) = x2, above the x-axis, andbetween the vertical lines x = 2 and x = 6.



Quick Quiz1. If {s1, s2, s3, s4} 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 s3 ? The largest?2. What are the lower and upper Riemann sums for f(x) = x2 when the order 4 uniform partition of [−1, 2]is used?3. Evaluate

4. True or false:

5. True or false:

Chapter 5-The Integral5.3 Rules for Integration


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



EXAMPLE: Calculate the integral

EXAMPLE: Suppose that g is a continuous function on the interval [4, 9] that satisfies

Calculate



Reversing the Direction of Integration

EXAMPLE: Evaluate



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



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



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



The Mean Value Theorem for Integrals

THEOREM: Let f be continuous on [a,b]. There is a value c in (a,b) such that



Quick Quiz

1. What is

2. Calculate

3. If and then what is

4. What is the average value of f(t) = t2 for 1 ≤t ≤ 4?

Chapter 5-The Integral5.4 The Fundamental Theorem of Calculus


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



Examples Illustrating the First Part of the Fundamental Theorem

EXAMPLE: Compute

EXAMPLE: Calculate



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?



Examples Illustrating the Second Part of the Fundamental Theorem

EXAMPLE: Calculate the derivative of

with respect to x.



Quick Quiz

1. Suppose that What is F’(2)?



4. Evaluate

Chapter 5-The Integral5.5 A Calculus Approach to the Logarithm and Exponential Functions




THEOREM: The natural logarithm has the following properties:



Properties of the Natural LogarithmTHEOREM: Let x and y be positive and p be any number. Then



Graphing the Natural Logarithm

THEOREM: The natural logarithm is an increasing function with domain equal to the set of positive realnumbers. 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.



The Exponential Function

We define b = exp (a)

if and only if a=ln(b)



Properties of the Exponential Function

• exp(s+t) = exp(s)exp(t)• exp(s-t) = exp(s)/exp(t)• (exp(s))t = exp(st)



Derivatives and Integrals Involving the Exponential Function

THEOREM: The exponential function satisfies



The Number e

Define e = exp(1)

so

ln( e ) = 1



Logarithms and Powers with Arbitrary Bases

ax = exp(xln(a))

EXAMPLE: Use the above formula to derive the Power Rule of differentiation



Logarithms and Powers with Arbitrary Bases

THEOREM: If a,b>0 and x,y are real numbers, then



Logarithms with Arbitrary Bases

THEOREM: For any fixed positive a ≠ 1, the function xax has domain of the reals and image of positive reals. The function x loga(x) has domain positive reals and image reals. The two functions are inverses of each other.



Logarithms with Arbitrary Bases

THEOREM: If x, y, a>0 and a≠ 1, then



Differentiation and Integration of ax and loga(x)



Quick Quiz1. 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 ax = eu?

4. Suppose that a is a positive constant not equal to 1. For what value of k is loga (x) = k · ln(x)?

Chapter 5-The Integral5.6 Integration by Substitution


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) dx3. 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.



Some Examples of Indefinite Integration by Substitution

EXAMPLE: Evaluate the indefinite integral sin4(x) cos(x) dx.

EXAMPLE: Compute the integral



The Method of Substitution for Definite Integrals

EXAMPLE: Evaluate the integral

EXAMPLE: Calculate



When an Integration Problem Seems to Have Two Solutions

EXAMPLE: Compute the indefinite integral sin(x)cos(x) dx.



Integral Tables

EXAMPLE: Use integral tables to compute



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



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 Integral5.7 More on the Calculation of Area


Rules for Calculating Area

(i) If f(x) ≥0 for x in [a, b], thenequals 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], thenequals the negative of the area of the region that is abovethe graph of f, below the x-axis, and between x = a and x = b.



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?



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



EXAMPLE: Calculate the area A of the region between the curves f(x) = −x2 + 6 and g(x) = 3x2 − 8 for xin the interval [−1, 1].

The Area between Two Curves



EXAMPLE: Compute the area between the curves x = y2 − y − 4 and x = −y2 + 3y + 12.

Reversing the Roles of the Axes



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−y2?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 Integral5.8 Numerical Techniques of Integration


The Midpoint RuleLet 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 MN be the midpoint approximation

If C is a constant such that |f’’(x)| ≤C for a ≤x ≤b, then



The Midpoint Rule

EXAMPLE: Let f(x) = 1/(1+x2). Estimate using the Midpoint Rule with N=4. Based on this approximation, how small might the exact value of A be? How large?



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/(12N2).



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



Simpson’s Rule

EXAMPLE: Let f(x) = 1/(1 + x2). Estimate using Simpson’s Rule with N =4. In what interval of real numbers does A lie?



Quick Quiz

1. Approximate by the Midpoint Rule with N = 2.

2. Approximate by the Trapezoidal Rule with N = 2.

3. Approximate by Simpson’s Rule with N = 4.

4. 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.

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