. Engineering education : proceedings of the ... annual meeting of the Society for the Promotion of Engineering Education . the above ex-pansion. The area of the third term is {kAx^)/S, that of thefourth one is {kAx^)/4: and so on. When we consider fxdx=^x-/2 we are accustomed to thinkthat x^/2 is the sum of a great number of terms of the formof xdx; it may as well be regarded as the sum of one or moreterms of the form xAx -f- (Aa:;-/2). To justify these statements and illustrate their meaning we 322 THE MEANING OF INTEGRATION IN CALCULUS. will now consider a few simple examples, but before do
. Engineering education : proceedings of the ... annual meeting of the Society for the Promotion of Engineering Education . the above ex-pansion. The area of the third term is {kAx^)/S, that of thefourth one is {kAx^)/4: and so on. When we consider fxdx=^x-/2 we are accustomed to thinkthat x^/2 is the sum of a great number of terms of the formof xdx; it may as well be regarded as the sum of one or moreterms of the form xAx -f- (Aa:;-/2). To justify these statements and illustrate their meaning we 322 THE MEANING OF INTEGRATION IN CALCULUS. will now consider a few simple examples, but before doing thisit is advisable to define one term which we will use and tostate the meaning which we propose to give to two commonexpressions of calculus. The complete differential of a function is the entire incre-ment of the function. The complete differential of x^ isSx-^x 4- 3xAa;2 + AxK The differential coefficient is the coefficient of the first powerof Arc in the complete differential. The integral of f{x)I^x is that function of x for which f{x)is the coefficient of the first power of Ax in the complete Fig. 2. These definitions require no change in the methods of in-tegration but merely give a different interpretation to theoperation. In many cases integration is a mere algebraicprocess. Important applications can be made which involvein no way the ideas of limits or infinitesimals, and many morecan be made which require the use of the limit of an infiniteseries but do not require that the differentials be regardedas infinitesimal. THE MEANING OF INTEGRATION IN CALCULUS. 323 Fig. 2 is a triangle of which the base is equal to the will find the area, taking the vertex as the origin and theX axis perpendicular to the base. An element is made up ofthe parallelogram of area icAa; and the shaded triangle ofbase and altitude Ax which is similar to the entire triangleand to the triangle of base and altitude x. If A representsarea A = fx^x. We recognize xAx
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