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MATH 318: MATHEMATICAL ANALYSIS II Chapter 7: INTEGRAL CALCULUS

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1 MATH 318: MATHEMATICAL ANALYSIS II Chapter 7: INTEGRAL CALCULUS
ISHIK UNIVERSITY FACULTY OF EDUCATION Mathematics Education Department MATH 318: MATHEMATICAL ANALYSIS II Chapter 7: INTEGRAL CALCULUS ORHAN TUĞ (PhDc) Mathematics Education Department

2 In the previous chapter we used the tangent and velocity problems to introduce the derivative, which is the central idea in differential calculus. In much the same way, this chapter starts with the area and distance problems and uses them to formulate the idea of a definite integral, which is the basic concept of integral calculus. There is a connection between integral calculus and differential calculus. The Fundamental Theorem of Calculus relates the integral to the derivative, and we will see in this chapter that it greatly simplifies the solution of many problems. Mathematics Education Department

3 The Area Problem Find the area of the following region:
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4 Mathematics Education Department

5 Mathematics Education Department

6 Mathematics Education Department

7 Mathematics Education Department

8 The Definite Integral Mathematics Education Department

9 Mathematics Education Department

10 Mathematics Education Department

11 Evaluating Integrals Mathematics Education Department

12 Properties of the Definite Integral
1: 2: 3: 4: 5: 6: Mathematics Education Department

13 7: Mathematics Education Department

14 8: 9: 10: 11: Mathematics Education Department

15 12: Mathematics Education Department

16 The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus is appropriately named because it establishes a connection between the two branches of calculus: differential calculus and integral calculus. Differential calculus arose from the tangent problem, whereas integral calculus arose from a seemingly unrelated problem, the area problem. Newton’s teacher at Cambridge, Isaac Barrow (1630–1677), discovered that these two problems are actually closely related. In fact, he realized that differentiation and integration are inverse processes. The Fundamental Theorem of Calculus gives the precise inverse relationship between the derivative and the integral. It was Newton and Leibniz who exploited this relationship and used it to develop calculus into a systematic mathematical method. In particular, they saw that the Fundamental Theorem enabled them to compute areas and integrals very easily without having to compute them as limits of sums Mathematics Education Department

17 Mathematics Education Department

18 Mathematics Education Department

19 Mathematics Education Department

20 Differentiation and Integration as Inverse Processes
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21 The Fundamental Theorem of Calculus
Importance of The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus is unquestionably the most important theorem in calculus and, indeed, it ranks as one of the great accomplishments of the human mind. Before it was discovered, from the time of Eudoxus and Archimedes to the time of Galileo and Fermat, problems of finding areas, volumes, and lengths of curves were so difficult that only a genius could meet the challenge. But now, armed with the systematic method that Newton and Leibniz fashioned out of the Fundamental Theorem, we will see in the chapters to come that these challenging problems are accessible to all of us. Mathematics Education Department

22 Indefinite Integrals or Antiderivatives
You should distinguish carefully between definite and indefinite integrals. A definite integral is a number, whereas an indefinite integral is a function (or family of functions). Mathematics Education Department

23 Table of Indefinite Integrals
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24 Applications of The Net Change Theorem
The integral of a rate of change is the net change:: Mathematics Education Department

25 Mathematics Education Department

26 Mathematics Education Department

27 Substitution Rule The Substitution Rule
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28 Symmetry in Definite Integral
Integrals of Symmetric Functions Mathematics Education Department

29 The Logarithm Defined as an Integral
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30 Laws of Logarithms Definition: Definition:
The general logarithmic function with base is the function defined by Mathematics Education Department

31 Mathematics Education Department

32 The Exponential Function
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33 Definition: Mathematics Education Department

34 Mathematics Education Department

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