592 VIII. ON THE STRUCTURE OF MATHEMATICS |
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In the above notes we have not attempted to give the reader more than some structural and methodological notions, and what amount really to short structural explanations of definitions which will be useful later on. The reader can find many excellent books which give all the additional information he may want.
Section C. On the integral calculus.So far, we have been studying a method by which to find the variation of a given function corresponding to an indefinitely small variation of our variable. We saw that the
rate of change of our function was given by the first derivative, which in turn was also a function (usually different) of our independent variable and so could itself vary and have a rate of change, and so give us a second derivative , .And now we must explain briefly the inverse problem; namely, given the derivative to find the function. In symbols, givenfind
U.The function
U is called the integral of « with respect to x, or, in symbols,To integrate a function
f{x) is to find a function F(x) which when differentiated gives again the function f(x) with which we started. As in this work we are not interested in computations, but only in the structural, methodological, and semantic aspects, the inverse problem of differentiation; namely, integration, is less important for us here, and I will explain only a single example. We have already differentiated the functionandfound its derivative . Just as the derivative of the sum of a
number of functions is equal to the sum of their derivatives, a similar rule holds for the integrals; namely, that the integral of the sum of a number of functions is equal to the sum of their integrals. Hence we can take in our example only the first term of our equation. In symbols
D±{2x=6x^{3}) ^{2}; in words, the derivative ofIn a problem in integration we would have
6xgiven and we would have to find the original function from which ^{2} 6xwas obtained by differentiation. In our case the solution is already given; namely, In general^{i} the solution of problems of integration is largely dependent on the ingenuity of the solver, although we have a number of standard formulae and methods. The geometrical meaning of integration is much more interesting for us and we will give a short explanation of it.
If we consider the curve given by an equation and the area bounded by the
X axis, the two ordinates whose abscissas are and x=b and the curve, we may find the area as follows: |
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♦The constant of integration is omitted so as not to confuse the reader. |
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