DIFFERENTIAL AND INTEGRAL CALCULUS. 19 



then if U denote the area aAMP measured from any fixed 

 ordinate a A, then Z7+A?7=area aANQ, or A 7= area 

 PMNQ, which for all forms of the curve, provided < (x) 

 is not infinite, will lie between the areas of the two rect- 

 angles PA 7 , MQ or between $ (x) . Aa? and (f> (x -f Aa?) . Aa? ; 



hence - lies between $ (a?) and < (a; -f Aic) ; or 

 Aa? 



Now suppose i/r (a?) to be a function of a?, such that 

 TT' x = (> a; then 



T' ( x ) = (f> (a;), then 



i.e. U=ijr (x) + C the indefinite integral. 



The original differential equation is obviously true 

 whatever be the position of aA, and hence the necessity 

 of an undetermined constant in the integral. As soon 

 as the initial ordinate Aa is chosen this constant is de- 

 termined ; for if OA = a, then when x = a, U= 0, or 

 = tjr (a) + Cj or U=ty (x) i|r (a] the corrected integral. 

 If the final ordinate be Bb, where x = bj we shall have 

 i/r (b) ^r (a) the definite integral, which is written, before 



f 6 

 determination, as I c/> (x) dx. 



J a / 



The meaning of the symbols /$ (x} dx is the limit of 

 the sum of an infinite number of magnitudes such as 

 <f)(x).Ax when Aa? is made indefinitely small, i.e. in the 

 figure of such quantities as the rectangle PJV, for if the 

 whole base AB be divided into n equal parts, each = Aa?, 

 and on each be completed rectangles as PA 7 , MQ, the 

 sum of all the internal rectangles as PN will be less than 

 the area of the curve aABb, and the sum of all the ex- 

 ternal rectangles as MQ will be greater 5 but the sum of 

 the former or 2 }< (x) Aa?} is 



+ (f> (a 4- Ax) -f <f> ( + 2 Aa;) 4 . . .+ < {a -f (n - 1) Aa?J], 



