20 DIFFERENTIAL AND INTEGRAL CALCULUS. 



and of the latter or 2 {( (x + Ax) . Ax} is 



Ax {0 (a + Ax) + < (a + 2Ax) + . . .+ < (a 4 rc 

 a + ?i . Ax being = #, 



and the difference of these sums is Ax {(j> (a + raAx) < (a)} 

 or Ax {( (J) <f> (a)}, which vanishes when Ax = 0; hence 

 the area of the curve lying between these, which in the 

 limit coincide, will be equal to the limit of either, or 



ty(b}-ty (a) = the limit of Ax [< (a) + < (a + Ax) 



+ <(+ 2 Ax) +...+ </> }a + (rc- 1) Ax}] when Ax = 0, 



and each element of which this sum is composed is of the 

 form (/> (x) Ax, x having successively all values between 

 a and b when Ax is made indefinitely small ; the whole 

 of which facts are succinctly and conveniently recorded by 



the notation I <(x).dx, the / being originally only the 



J a 



long 8, and being used as being the initial letter of the 

 word sum. 



Every differential coefficient, which has been inves- 

 tigated, gives rise to a corresponding integral: 



(1) ( x m ) = mx m -* ; or jW"-Wx = x'", or Jx^dx = - ; 



CtX fit 



but to apply this in integration it is more convenient to 



x m+1 

 put m + 1 for r/?, so that /x"Wx = -- - + C. 



(In simply transforming differential coefficients we shall 

 omit the 0, but it must on no account be omitted in any 

 application). 



d . 1 Cdx 



(2) 



Here it may be asked why this integral is not deduced 

 at once as a particular case of (1) by putting m = 1, 



since then x m becomes - . It will be seen that in that 

 x 



x nHl 



case - , becomes GO for all finite values of x, whence 

 m+1 



