1. INTRODUCTORY MATHEMATICAL ANALYSIS For Business, Economics, and the Life and Social Sciences  2007 Pearson Education Asia Chapter 14 Integration

2.  2007 Pearson Education Asia INTRODUCTORY MATHEMATICAL ANALYSIS 0.Review of Algebra 1.Applications and More Algebra 2.Functions and Graphs 3.Lines, Parabolas, and Systems 4.Exponential and Logarithmic Functions 5.Mathematics of Finance 6.Matrix Algebra 7.Linear Programming 8.Introduction to Probability and Statistics

3.  2007 Pearson Education Asia 9.Additional Topics in Probability 10.Limits and Continuity 11.Differentiation 12.Additional Differentiation Topics 13.Curve Sketching 14.Integration 15.Methods and Applications of Integration 16.Continuous Random Variables 17.Multivariable Calculus INTRODUCTORY MATHEMATICAL ANALYSIS

4.  2007 Pearson Education Asia To define the differential. To define the anti-derivative and the indefinite integral. To evaluate constants of integration. To apply the formulas for. To handle more challenging integration problems. To evaluate simple definite integrals. To apply Fundamental Theorem of Integral Calculus. Chapter 14: Integration Chapter Objectives

5.  2007 Pearson Education Asia To use Trapezoidal rule or Simpson’s rule. To use definite integral to find the area of the region. To find the area of a region bounded by two or more curves. To develop concepts of consumers’ surplus and producers’ surplus. Chapter 14: Integration Chapter Objectives

6.  2007 Pearson Education Asia Differentials The Indefinite Integral Integration with Initial Conditions More Integration Formulas Techniques of Integration The Definite Integral The Fundamental Theorem of Integral Calculus 14.1) 14.2) 14.3) Chapter 14: Integration Chapter Outline 14.4) 14.5) 14.6) 14.7)

7.  2007 Pearson Education Asia Approximate Integration Area Area between Curves Consumers’ and Producers’ Surplus 14.8) 14.9) 14.10) Chapter 14: Integration Chapter Outline 14.11)

8.  2007 Pearson Education Asia Chapter 14: Integration 14.1 Differentials Example 1 – Computing a Differential The differential of y, denoted dy or d(f(x)), is given by Find the differential of and evaluate it when x = 1 and ∆x = 0.04. Solution: The differential is When x = 1 and ∆x = 0.04,

9.  2007 Pearson Education Asia Chapter 14: Integration 14.1 Differentials Example 3 - Using the Differential to Estimate a Change in a Quantity A governmental health agency examined the records of a group of individuals who were hospitalized with a particular illness. It was found that the total proportion P that are discharged at the end of t days of hospitalization is given by Use differentials to approximate the change in the proportion discharged if t changes from 300 to 305.

10.  2007 Pearson Education Asia Chapter 14: Integration 14.1 Differentials Example 3 - Using the Differential to Estimate a Change in a Quantity Example 5 - Finding dp/dq from dq/dp Solution: We approximate ∆P by dP, Solution:

11.  2007 Pearson Education Asia Chapter 14: Integration 14.2 The Infinite Integral An antiderivative of a function f is a function F such that. In differential notation,. Integration states that Basic Integration Properties:

12.  2007 Pearson Education Asia Chapter 14: Integration 14.2 The Infinite Integral Example 1 - Finding an Indefinite Integral Example 3 - Indefinite Integral of a Constant Times a Function Example 5 - Finding Indefinite Integrals Find. Solution: Find. Solution:

13.  2007 Pearson Education Asia Find. Solution: Chapter 14: Integration 14.2 The Infinite Integral Example 7 - Indefinite Integral of a Sum and Difference

14.  2007 Pearson Education Asia Find Solution: Chapter 14: Integration 14.2 The Infinite Integral Example 9 - Using Algebraic Manipulation to Find an Indefinite Integral

15.  2007 Pearson Education Asia Chapter 14: Integration 14.3 Integration with Initial Conditions Example 1 - Initial-Condition Problem Use initial conditions to find the constant, C. If y is a function of x such that y’ = 8x − 4 and y(2) = 5, find y. Solution: We find the integral, Using the condition, The equation is

16.  2007 Pearson Education Asia Chapter 14: Integration 14.3 Integration with Initial Conditions Example 3 - Income and Education For a particular urban group, sociologists studied the current average yearly income y (in dollars) that a person can expect to receive with x years of education before seeking regular employment. They estimated that the rate at which income changes with respect to education is given by where y = 28,720 when x = 9. Find y.

17.  2007 Pearson Education Asia Chapter 14: Integration 14.3 Integration with Initial Conditions Example 3 - Income and Education Solution: We have When x = 9, Therefore,

18.  2007 Pearson Education Asia Chapter 14: Integration 14.3 Integration with Initial Conditions Example 5 - Finding Cost from Marginal Cost In the manufacture of a product, fixed costs per week are $4000. (Fixed costs are costs, such as rent and insurance, that remain constant at all levels of production during a given time period.) If the marginal-cost function is where c is the total cost (in dollars) of producing q pounds of product per week, find the cost of producing 10,000 lb in 1 week.

19.  2007 Pearson Education Asia Chapter 14: Integration 14.3 Integration with Initial Conditions Example 5 - Finding Cost from Marginal Cost Solution: The total cost c is When q = 0, c = 4000. Cost of 10,000 lb in one week,

20.  2007 Pearson Education Asia Chapter 14: Integration 14.4 More Integration Formulas Power Rule for Integration Integrating Natural Exponential Functions Integrals Involving Logarithmic Functions

21.  2007 Pearson Education Asia Chapter 14: Integration 14.4 More Integration Formulas Basic Integration Formulas

22.  2007 Pearson Education Asia Chapter 14: Integration 14.4 More Integration Formulas Example 1 - Applying the Power Rule for Integration Find the integral of Solution:

23.  2007 Pearson Education Asia Chapter 14: Integration 14.4 More Integration Formulas Example 3 - Adjusting for du Find Solution:

24.  2007 Pearson Education Asia Chapter 14: Integration 14.4 More Integration Formulas Example 5 - Integrals Involving Exponential Functions Find Solution:

25.  2007 Pearson Education Asia Chapter 14: Integration 14.4 More Integration Formulas Example 7 - Integrals Involving Exponential Functions Find Solution:

26.  2007 Pearson Education Asia Chapter 14: Integration 14.5 Techniques of Integration Example 1 - Preliminary Division before Integration Find

27.  2007 Pearson Education Asia Chapter 14: Integration 14.5 Techniques of Integration Example 3 - An Integral Involving b u Find Solution: General formula for integrating b u is

28.  2007 Pearson Education Asia Chapter 14: Integration 14.6 The Definite Integral Example 1 - Computing an Area by Using Right-Hand Endpoints For area under the graph from limit a  b, x is called the variable of integration and f (x) is the integrand. Find the area of the region in the first quadrant bounded by f(x) = 4 − x 2 and the lines x = 0 and y = 0. Solution: Since the length of [0, 2] is 2, ∆x = 2/n.

29.  2007 Pearson Education Asia Chapter 14: Integration 14.6 The Definite Integral Example 1 - Computing an Area by Using Right-Hand Endpoints Summing the areas, we get We take the limit of S n as n→∞: Hence, the area of the region is 16/3.

30.  2007 Pearson Education Asia Chapter 14: Integration 14.6 The Definite Integral Example 3 - Integrating a Function over an Interval Integrate f (x) = x − 5 from x = 0 to x = 3. Solution:

31.  2007 Pearson Education Asia Chapter 14: Integration 14.7 The Fundamental Theorem of Integral Calculus Fundamental Theorem of Integral Calculus If f is continuous on the interval [a, b] and F is any antiderivative of f on [a, b], then Properties of the Definite Integral If a > b, then If limits are equal,

32.  2007 Pearson Education Asia Chapter 14: Integration 14.7 The Fundamental Theorem of Integral Calculus Properties of the Definite Integral 1. is the area bounded by the graph f(x). 2. 3. 4. 5.

33.  2007 Pearson Education Asia Chapter 14: Integration 14.7 The Fundamental Theorem of Integral Calculus Example 1 - Applying the Fundamental Theorem Find Solution:

34.  2007 Pearson Education Asia Chapter 14: Integration 14.7 The Fundamental Theorem of Integral Calculus Example 3 - Evaluating Definite Integrals Find Solution:

35.  2007 Pearson Education Asia Chapter 14: Integration 14.7 The Fundamental Theorem of Integral Calculus Example 5 - Finding a Change in Function Values by Definite Integration The Definite Integral of a Derivative The Fundamental Theorem states that A manufacturer’s marginal-cost function is. If production is presently set at q = 80 units per week, how much more would it cost to increase production to 100 units per week? Solution: The rate of change of c is dc/dq is

36.  2007 Pearson Education Asia Chapter 14: Integration 14.8 Approximate Integration Trapezoidal Rule To find the area of a trapezoidal area, we have

37.  2007 Pearson Education Asia Chapter 14: Integration 14.8 Approximate Integration Example 1 - Trapezoidal Rule Use the trapezoidal rule to estimate the value of for n = 5. Compute each term to four decimal places, and round the answer to three decimal places. Solution: With n = 5, a = 0, and b = 1,

38.  2007 Pearson Education Asia Chapter 14: Integration 14.8 Approximate Integration Example 1 - Trapezoidal Rule Solution: The terms to be added are Estimate for the integral is

39.  2007 Pearson Education Asia Chapter 14: Integration 14.8 Approximate Integration Simpson’s Rule Approximating the graph of f by parabolic segments gives

40.  2007 Pearson Education Asia Chapter 14: Integration 14.8 Approximate Integration Example 3 - Demography A function often used in demography (the study of births, marriages, mortality, etc., in a population) is the life-table function, denoted l. In a population having 100,000 births in any year of time, l(x) represents the number of persons who reach the age of x in any year of time. For example, if l(20) = 98,857, then the number of persons who attain age 20 in any year of time is 98,857.

41.  2007 Pearson Education Asia Chapter 14: Integration 14.8 Approximate Integration Example 3 - Demography Suppose that the function l applies to all people born over an extended period of time. It can be shown that, at any time, the expected number of persons in the population between the exact ages of x and x + m, inclusive, is given by The following table gives values of l(x) for males and females in the United States. Approximate the number of women in the 20–35 age group by using the trapezoidal rule with n = 3.

42.  2007 Pearson Education Asia Chapter 14: Integration 14.8 Approximate Integration Example 3 - Demography Life table:

43.  2007 Pearson Education Asia Chapter 14: Integration 14.8 Approximate Integration Example 3 - Demography Solution: We want to estimate Thus The terms to be added are By the trapezoidal rule,

44.  2007 Pearson Education Asia Chapter 14: Integration 14.9 Area Example 1 - Using the Definite Integral to Find Area The width of the vertical element is ∆x. The height is the y-value of the curve. The area is defined as Find the area of the region bounded by the curve and the x-axis.

45.  2007 Pearson Education Asia Chapter 14: Integration 14.9 Area Example 1 - Using the Definite Integral to Find Area Solution: Summing the areas of all such elements from x = −3 to x = 2,

46.  2007 Pearson Education Asia Chapter 14: Integration 14.9 Area Example 3 - Finding the Area of a Region Find the area of the region between the curve y = e x and the x-axis from x = 1 to x = 2. Solution: We have

47.  2007 Pearson Education Asia Chapter 14: Integration 14.9 Area Example 5 - Statistics Application In statistics, a (probability) density function f of a variable x, where x assumes all values in the interval [a, b], has the following properties: The probability that x assumes a value between c and d, which is written P(c ≤ x ≤ d), where a ≤ c ≤ d ≤ b, is represented by the area of the region bounded by the graph of f and the x-axis between x = c and x = d.

48.  2007 Pearson Education Asia Chapter 14: Integration 14.9 Area Example 5 - Statistics Application Hence For the density function f(x) = 6(x − x2), where 0 ≤ x ≤ 1, find each of the following probabilities.

49.  2007 Pearson Education Asia Chapter 14: Integration 14.9 Area Example 5 - Statistics Application Solution: a. b.

50.  2007 Pearson Education Asia Chapter 14: Integration 14.10 Area between Curves Example 1 - Finding an Area between Two Curves Vertical Elements The area of the element is Find the area of the region bounded by the curves y = √x and y = x. Solution: Eliminating y by substitution,

51.  2007 Pearson Education Asia Chapter 14: Integration 14.10 Area between Curves Example 3 - Area of a Region Having Two Different Upper Curves Find the area of the region between the curves y = 9 − x 2 and y = x 2 + 1 from x = 0 to x = 3. Solution: The curves intersect when

52.  2007 Pearson Education Asia Chapter 14: Integration 14.10 Area between Curves Example 5 - Advantage of Horizontal Elements Find the area of the region bounded by the graphs of y 2 = x and x − y = 2. Solution: The intersection points are (1,−1) and (4, 2). The total area is

53.  2007 Pearson Education Asia Chapter 14: Integration 14.11 Consumers’ and Producers’ Surplus Example 1 - Finding Consumers’ Surplus and Producers’ Surplus Consumers’ surplus, CS, is defined as Producers’ surplus, PS, is defined as The demand function for a product is where p is the price per unit (in dollars) for q units. The supply function is. Determine consumers’ surplus and producers’ surplus under market equilibrium.

54.  2007 Pearson Education Asia Chapter 14: Integration 14.11 Consumers’ and Producers’ Surplus Example 1 - Finding Consumers’ Surplus and Producers’ Surplus Solution: Find the equilibrium point (p 0, q 0 ), Consumers’ surplus is Producers’ surplus is