QABE_Notes17 - QABE Lecture 17 Integral Calculus: Unlocking...

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Unformatted text preview: QABE Lecture 17 Integral Calculus: Unlocking Economic Dynamics School of Economics, UNSW 2011 Contents 1 Introduction 1 2 Why Integration? 2 3 The Indefinite Integral 3 3.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.2 The Constant of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.3 Tools of the Trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.4 Techniques of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4 The Definite Integral 7 4.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.2 As an Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1 Introduction In the previous two lectures, we have been interested in finding the rate of change of some function, and we have used this to (for example) identify where some output is at a maximum or minimum. While this is very useful for a variety of applications, it seems natural to expect that there will be times where rather than having access to some primitive function describing how an output is affected by an input, we will instead have observed the rate of change itself, and so wish to find the primitive function. In fact, this is the other side of the coin of our differentiation work of the past two lectures, and it involves, appropriately enough, finding the antiderivative of a function, or as it is commonly known, the integral . In particular, integration (the process of going from the derivative to its primitive) has a natural application when we deal with quantities that change over time. That is, as in our first example below, we might observe the birth and death rate of a nations population, and wish to determine what the population will actually be at some time point in the future. This is exactly the kind of step that integration can deliver if we can put an equation down for the rate of change of the population, then we should be able to go back one step to an equation of population at a given time. Notice the word should in the previous sentence. Two things are against us. First, as with differentiation, there are integration rules for some functions, but not all! Second, 1 ECON 1202/ECON 2291: QABE c School of Economics, UNSW even if we do have a rule that will give us the primitive function we are after, we still need one more piece of information a single measurement of the primitive function at a...
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This note was uploaded on 09/23/2011 for the course ECON 1202 taught by Professor Lorettiisabelladobrescu during the One '11 term at University of New South Wales.

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QABE_Notes17 - QABE Lecture 17 Integral Calculus: Unlocking...

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