340 MATHEMATICS. 



xy is x.dy -f y.dx : that of x* is 2 x . dx ; and that of x 3 is 3x* dx. 

 In general, the differential of a product, is found by multiplying the 

 differential of each variable by the product of all the other variables, 

 and taking the sura of these several products. The differential of 

 x_ . g y.dx x.dy ; ^ differential of the sine of x is cos x .dx: the 



y y* 



differential of the cosine of x is sin x. dx : and that of the loga- 

 rithm of x is . By differentiating anew the first differential equa- 



tion, we obtain the second differential equation, and from this the 

 third, and higher equations ; which are necessary in the application 

 of Taylor's and Maclaurin's theorems : but these, we have no room 

 here to explain. 



2. The object of the Integral Calculus, is, having given the 

 differential coefficient of any function, to find the function itself. 

 Hence, it is the reverse of the Differential Calculus ; and was called 

 by Newton the Inverse Method of Fluxions. In this view, the func- 

 tion is called the integral, or fluent ; being considered as the sum 

 of all the successive increments which together make up the function 

 sought. To find the integral of any expression, is to find the quan- 

 tity which will have that expression for its differential, or differential 

 coefficient. Thus, the differential of a x -f h being a.dx, conversely 

 we say that the integral of a.dx is ax -f- b. The integral of any 



quantity, is designated by writing before it the character /, resem- 



bling the letter s, the initial of the word sum ; as d was used to de- 

 signate the differential, by a similar alliteration. Thus we have 

 d (ax -f- b] == dy = a.dx; and 



J a.dx = y ax -f- b. 



The constant term b, was called by Newton the correction of the 

 fluent: and it cannot be found immediately by the integration, be- 

 cause this process only gives the variable terms, of which this constant 

 term is entirely independent. 



This explanation of the Integral Calculus, will serve to show how 

 it may be applied to the rectification of a curve, that is, the finding 

 of its length ; or to the quadrature of a surface, that is, the finding of 

 its area ; or to the cubature of a solid, by which we measure its so- 

 lidity. Taking, for example, the quadrature of the parabola, the 

 equation of which is i/ 3 = 2 px ; (p. 336) ; and calling s the area com- 

 prehended between the axis, the curve, and a given ordinate y, (or 

 PH, Plate VII., Fig. 18,) we assume the equation ds = y.dx ; each 

 member expressing the infinitely small area comprehended between 

 two consecutive ordinates : and substituting, in this, the value of dx, 

 found from the differential equation, 2y . dy = 2p . dx, which gives 



ds = ~ ; the inte S ral of 



y* 



~ ' 



integral requires no correction, or constant term, be- 

 cause it already makes the area s = when y becomes zero, which 



