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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 Alevel 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 noninteger 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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 Spring '10
 Vines
 Economics

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