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Chapter 15 Methods and Applications of Integration

Chapter 15 Methods and Applications of Integration. Chapter 15: Methods and Applications of Integration Chapter Objectives. 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.

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Chapter 15 Methods and Applications of Integration

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  1. Chapter 15 Methods and Applications of Integration

  2. Chapter 15: Methods and Applications of Integration • Chapter Objectives • 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.

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

  4. 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,

  5. 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,

  6. Chapter 15: Methods and Applications of Integration • 15.1 Integration by Parts • Example 5 – Applying Integration by Parts Twice Determine Solution: Let and Thus,

  7. Chapter 15: Methods and Applications of Integration • 15.1 Integration by Parts • Example 5 – Applying Integration by Parts Twice Solution (cont’d):

  8. 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,

  9. 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,

  10. 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,

  11. 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,

  12. 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.

  13. 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,

  14. 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 a2 = 2, we have du = 2 dx. Thus,

  15. 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 functionf (x) is given by Find the average value of the function f(x)=x2over the interval [1, 2]. Solution:

  16. 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,

  17. Chapter 15: Methods and Applications of Integration • Example 1 – Separation of Variables Solution (cont’d):

  18. 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,

  19. 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

  20. 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?

  21. 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,

  22. 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 andT(t) = temperature of the body at time t. By Newton’s law of cooling,

  23. Chapter 15: Methods and Applications of Integration • 15.6 More Applications of Differential Equations • Example 3 – Time of Murder Solution (cont’d):

  24. 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.

  25. Chapter 15: Methods and Applications of Integration • 15.7 Improper Integrals • The improper integral is defined as • The improper integral is defined as

  26. 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.

  27. 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:

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