Integration - CHAPTER 10 Integration Integration can be...

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Unformatted text preview: CHAPTER 10 Integration Integration can be thought of as the opposite of differentiation but is also a method for finding the area under a graph . It is an important mathematical technique, which will be familiar if you have done A-level maths. In this chapter we look at techniques for integrating standard functions, including integration by substitution and by parts and at economic applications including calculation of consumer surplus . ./ 1. The Reverse of Differentiation If we have a function y ( x ), we know how to find its derivative dy dx by the process of differen- tiation . For example: y ( x ) = 3 x 2 + 4 x- 1 dy dx = 6 x + 4 Integration is the reverse process: When you know the derivative of a function, dy dx , the process of finding the original function, y , is called integration. For example: dy dx = 10 x- 3 y ( x ) = ? If you think about how differentiation works, you can probably see that the answer could be: y = 5 x 2- 3 x However, there are many other possibilities; it could be y = 5 x 2- 3 x +1, or y = 5 x 2- 3 x- 20, or ... in fact it could be any function of the form: y = 5 x 2- 3 x + c where c is a constant. We say that (5 x 2- 3 x + c ) is the integral of (10 x- 3) and write this as: The with integral respect of to x Z (10 x- 3) dx = 5 x 2- 3 x + c 173 174 10. INTEGRATION c is referred to as an arbitrary constant or a constant of integration. More generally: Integration is the reverse of differentiation. If f ( x ) is the derivative of a function f ( x ), then the integral of f ( x ) is f ( x ) (plus an arbitrary constant): Z f ( x ) dx = f ( x ) + c 1.1. Integrating Powers and Polynomials In the example above you can see that since differentiating powers of x involves reducing the power by 1, integrating powers of x must involve increasing the power by 1. The rule is: Integrating Powers of x : Z x n dx = 1 n + 1 x n +1 + c ( n 6 =- 1) It is easy to check that this rule works by differentiating: d dx 1 n + 1 x n +1 + c = x n You can also see from this that the rule doesnt work when n =- 1. But it works for other negative powers, for zero, and for non-integer powers see the examples below. We can apply this rule to integrate polynomials. For example: Z (4 x 2 + 6 x- 3) dx = Z (4 x 2 + 6 x- 3 x ) dx = 4 1 3 x 3 + 6 1 2 x 2- 3 x 1 + c = 4 3 x 3 + 3 x 2- 3 x + c It is easy to make mistakes when integrating. You should always check your answer by dif- ferentiating it to make sure that you obtain the original function. Examples 1.1 : Integrating Powers and Polynomials (i) What is the integral of x 4- 2 x + 5? Z ( x 4- 2 x + 5) dx = 1 5 x 5- x 2 + 5 x + c (ii) Integrate 2- t 5 5 . Z 2- t 5 5 dt = 2 t- t 6 30 + c 10. INTEGRATION 175 (iii) If dy dx = (2- x )(4- 3 x ), what is y ?...
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Integration - CHAPTER 10 Integration Integration can be...

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