UNDERSTANDING INTEGRATION

Transcription

1 UNDERSTANDING INTEGRATION Dear Reaer The concept of Integration, mathematically speaking, is the "Inverse" of the concept of result, the integration of, woul give us back the function f(). This, in a way, is similar to aition an subtraction or multiplication an ivision. Suppose we a 8 to 16, we get the result 24. If we now subtract 8 from 24, we get back the number 16. Similarly, if we multiply, say 9 by 6, we get 54. On iviing 54 by 6, we get back the number 9. Let us first unerstan the physical meaning of the term integration. Consier the mass of a tiny roplet of water. It woul be very small. Net, consier the mass of a collection of a very-very large number of such roplets. It woul, as we know, be a reasonable an measurable quantity. We are, in a way, here aing a very-very large number of quantities, each of which is very-very small. Integration is the mathematical technique that enables us to fin such sums- the sum of a very-very large number of quantities each of which is very-very small. We coul in a rather crue way, think of it as fining the prouct of (0) with (infinity). Mathematicians use a special symbol for integration, this symbol is. It is interesting to note that the symbol for integration, in a way, resembles the letter S that has been stretche out both ways. This may be linke with the physical meaning of integration- sum of a very-very large number of very-very small terms. The very very small terms, that are ae up through integration, are enote, mathematically, by the symbol f(). Here f() coul be any function of. For eample: we coul have f()= 2 +1 or f()= a sin or f()= 2 +log an so on. The aition of such small terms is known as integration an is epresse through the complete symbol. The symbol then implies the aition of a very very large number of very very small terms. 111

2 Let =, we woul then have / [f()] =. This is because the process of integration, an ifferentiation, are the "inverse" of each other. Mathematicians who evelope the techniques of integration; i.e. fining such sums, for ifferent kins of mathematical functions, foun that they coul carry out the relevant mathematical steps by thinking of integration as the inverse of ifferentiation. The integration of 0 It is important to remember that if y=constant say (c), the rate of change of y, woul have to be zero. We can, therefore, say that here 0. Remembering that integration is the inverse of ifferentiation, this result implies 0 = c=a constant. We can thus say The integration of 0, woul give us a constant as its result. This result will hol irrespective of the actual value of the constant (c). We, therefore say; The integration of 0, gives an ineterminate constant as its answer. This result has an interesting implication. Suppose we integrate a function f() an get as the answer. we woul then write = However we can always write f() = f()+0 But [ f()+0 ] = f() + (0) = + constant. This implies that instea of writing = we shoul write = + C 112

3 Therefore, the result of integration, of any given function, oes not give us a unique/efinite answer. To the result, we can always a a (arbitrary) constant. We now quote the mathematical results for integration of some stanar functions. Remember that the term, (+c), in these results, is ue to the above state, an iscusse, property of the integration of zero Integration of some common functions Derivatives Integrals (Anti erivatives) (i) Particularly, we note that (ii) (iii) (iv) (v) (vi) (vii) 11

4 [ *For case (i) For n= -1, we have n =.This function gives ln as its answer. Here (ln ) stans for logarithm of to a special base, known as e. It turns out that ln = 2.0 log, where log represents the logarithm of to the base 10.] Rules for Integration For the following statements an eamples, the symbols, a, b an c, all refer to terms that are constants Eample 1 114

5 Eample 2 Eample C Eample 4 Eample 5 115

6 Eample 6 =5 +4sec + C DO IT YOURSELF Fin the Following Integrals (a) (b) (c) () (e) (f) (g) (h) (i) (j) (k) ANSWERS 1. (a) 6 6 (b) 5 5 (c) () 116

7 (e) (f) 2 2 (g) (h) (i) (j) (k) 9sin 117