DIFFERENTIAL AND INTEGRAL CALCULUS. 33 



The result is obvious if we draw the two curves 

 y = $(x}, y = $(a x}) suppose BPQDG (fig. 7), and 

 bpqDc, then if OM=x, MP=4>(x}, Am = x, therefore 

 Om ax : then mp =MP, since mp = < (a Om] = (f>(x} MP. 



The area of the curve BOACl (p(x}dx, and that of 



* 



cOAb= I <j>(a x) dxj and these areas are manifestly equal, 



J o 



the bounding curves being similar and equal. 

 The following is also sometimes useful : 



for 



[ <t>(x)dx = I 



J o J o 



[ <j>(x)dx=^(2a) 

 ' 



= [ <f>(x)dx+ [ <f> (x) dx, 



J a *^0 



and if in the first term we put 2a z for X, it becomes 

 f c (2a-z)(-dz) or [ ^(2a-)<fo, or f <t>(2a-x}dx. 



J a J o J o 



This is also easily proved geometrically like the former. 



As examples of the use to be made of such formula, 

 take 



r-rr rir 



\ xF($'mx) dx= I (TT - x) F {sm(7r - x}} dx 



^o J o 



f-TT /"7T 



= TT I F(smx] dx I F(sm x} dx, 



J o ^o 



therefore I xF(smx)dx = I F(smx)dx. 



J o * 'o 



rr , f 71 " Xdx 7T [ v t~ 



Thus . = - 



J 1 + since 2 J 1 + 



, r v x sinxdx _ TT /""" sinccJic 

 J l+cos*aj" 2" J I 



+ cos 



1 / 



7T 



