INTRODUCTORY MATHEMATICAL ANALYSIS For Business, Economics, and the Life and Social Sciences  2011 Pearson Education, Inc. Chapter 15 Methods and Applications of Integration

 2011 Pearson Education, Inc. To develop and apply the formula for integration by parts. To show how to integrate a proper rational function. To illustrate the use of the table of integrals. To develop the concept of the average value of a function. To solve a differential equation by using the method of separation of variables. To develop the logistic function as a solution of a differential equation. To define and evaluate improper integrals. Chapter 15: Methods and Applications of Integration Chapter Objectives

 2011 Pearson Education, Inc. Integration by Parts Integration by Partial Fractions Integration by Tables Average Value of a Function Differential Equations More Applications of Differential Equations Improper Integrals 15.1) 15.2) 15.3) Chapter 15: Methods and Applications of Integration Chapter Outline 15.4) 15.5) 15.6) 15.7)

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.1 Integration by Parts Example 1 – Integration by Parts Formula for Integration by Parts Find by integration by parts. Solution: Let and Thus,

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.1 Integration by Parts Example 3 – Integration by Parts where u is the Entire Integrand Determine Solution: Let and Thus,

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.1 Integration by Parts Example 5 – Applying Integration by Parts Twice Determine Solution: Let and Thus,

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.1 Integration by Parts Example 5 – Applying Integration by Parts Twice Solution (cont’d):

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.2 Integration by Partial Fractions Example 1 – Distinct Linear Factors Express the integrand as partial fractions Determine by using partial fractions. Solution: Write the integral as Partial fractions: Thus,

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.2 Integration by Partial Fractions Example 3 – An Integral with a Distinct Irreducible Quadratic Factor Determine by using partial fractions. Solution: Partial fractions: Equating coefficients of like powers of x, we have Thus,

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.2 Integration by Partial Fractions Example 5 – An Integral Not Requiring Partial Fractions Find Solution: This integral has the form Thus,

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.3 Integration by Tables Example 1 – Integration by Tables In the examples, the formula numbers refer to the Table of Selected Integrals given in Appendix B of the book. Find Solution: Formula 7 states Thus,

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.3 Integration by Tables Example 3 – Integration by Tables Find Solution: Formula 28 states Let u = 4x and a = √3, then du = 4 dx.

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.3 Integration by Tables Example 5 – Integration by Tables Find Solution: Formula 42 states If we let u = 4x, then du = 4 dx. Hence,

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.3 Integration by Tables Example 7 – Finding a Definite Integral by Using Tables Evaluate Solution: Formula 32 states Letting u = 2x and a 2 = 2, we have du = 2 dx. Thus,

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.4 Average Value of a Function Example 1 – Average Value of a Function The average value of a function f (x) is given by Find the average value of the function f(x)=x 2 over the interval [1, 2]. Solution:

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.5 Differential Equations Example 1 – Separation of Variables We will use separation of variables to solve differential equations. Solve Solution: Writing y’ as dy/dx, separating variables and integrating,

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration Example 1 – Separation of Variables Solution (cont’d):

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.5 Differential Equations Example 3 – Finding the Decay Constant and Half-Life If 60% of a radioactive substance remains after 50 days, find the decay constant and the half-life of the element. Solution: Let N be the size of the population at time t,

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.6 More Applications of Differential Equations Logistic Function The function is called the logistic function or the Verhulst– Pearl logistic function. Alternative Form of Logistic Function

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.6 More Applications of Differential Equations Example 1 – Logistic Growth of Club Membership Suppose the membership in a new country club is to be a maximum of 800 persons, due to limitations of the physical plant. One year ago the initial membership was 50 persons, and now there are 200. Provided that enrollment follows a logistic function, how many members will there be three years from now?

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.6 More Applications of Differential Equations Example 1 – Logistic Growth of Club Membership Solution: Let N be the number of members enrolled in t years, Thus,

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.6 More Applications of Differential Equations Example 3 – Time of Murder A wealthy industrialist was found murdered in his home. Police arrived on the scene at 11:00 P.M. The temperature of the body at that time was 31◦C, and one hour later it was 30◦C. The temperature of the room in which the body was found was 22◦C. Estimate the time at which the murder occurred. Solution: Let t = no. of hours after the body was discovered and T(t) = temperature of the body at time t. By Newton’s law of cooling,

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.6 More Applications of Differential Equations Example 3 – Time of Murder Solution (cont’d):

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.6 More Applications of Differential Equations Example 3 - Time of Murder Solution (cont’d): Accordingly, the murder occurred about 4.34 hours before the time of discovery of the body (11:00 P.M.). The industrialist was murdered at about 6:40 P.M.

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.7 Improper Integrals The improper integral is defined as

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.7 Improper Integrals Example 1 – Improper Integrals Determine whether the following improper integrals are convergent or divergent. For any convergent integral, determine its value.

 2011 Pearson Education, Inc. Chapter 15: Methods and Applications of Integration 15.7 Improper Integrals Example 3 – Density Function In statistics, a function f is called a density function if f(x) ≥ 0 and. Suppose is a density function. Find k. Solution: