An elementary course of infinitesimal calculus . tuitively from the definition of Art. 86. For example, ra ra I (f)(x)dx = I ^{a — x)dx (1). This is proved by writing x = a — x, dx= — dx, 95-96] DEFINITE INTEGRALS. 233 the new limits of integration being « =a, x = 0, correspond-ing to x = 0,x = a, respectively. Thus ra ro ra I (a — ») dso = j ^(a — x) dx, Jo J a Jo the accent being dropped in the end, as no longer necessary. This process is equivalent to transferring the origin tothe point x = a, and reversing the direction of the axis of areas represented by the integrals in (1) are thu
An elementary course of infinitesimal calculus . tuitively from the definition of Art. 86. For example, ra ra I (f)(x)dx = I ^{a — x)dx (1). This is proved by writing x = a — x, dx= — dx, 95-96] DEFINITE INTEGRALS. 233 the new limits of integration being « =a, x = 0, correspond-ing to x = 0,x = a, respectively. Thus ra ro ra I (a — ») dso = j ^(a — x) dx, Jo J a Jo the accent being dropped in the end, as no longer necessary. This process is equivalent to transferring the origin tothe point x = a, and reversing the direction of the axis of areas represented by the integrals in (1) are thusseen to be identical. An important case of (1) is f ^/(sin 0) dO = f */(cos e)dd (2). Jo Jo Ex.\. Thus 1^sined6= Icos^ede. Hence each of these integrals = J ( (sin^6l + cos=6l)c?6l = J [^ dd = Jo Again, if ^ (x) be an even function of x, that is {-x) = 4>{x) (3), /« ra4>(x)dx = 2 4){x)dx (4),-a JO the area represented by the former integi-al being obviouslybisected by the axis of 234 INFINITESIMAL CALCULUS. [CH. VI On the other hand, if 0 (x) be an odd function of x, so that we have <li{-x) = -{x). r <^{x)dx = J ~a •(5)..(6), since in the sum, of which the definite integral is the limit(Art. 86), the element cf) (x) Sx is cancelled by the oppositely-signed element 0 (— x) 8x.
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