THE DEFINITE INTEGRAL 198 



tion is the converse of differentiation, it follows therefore, that in 

 order to obtain A, y must be integrated with respect to x. 



Then A - \y fa 



Considering the area KLMN, which is bounded by the axis 

 of x, the ordinates at x = a and x = b, and the portion KN of 

 the curve. Let this area be divided up into a number of thin 

 strips, the breadth of each strip being &r. 



The number of strips will be ^ 



The area of each strip will be approximately y Bx. 



Now, when &c is made very small, the number of strips taken is 

 considerably increased, the area of each strip is considerably 

 decreased, but at the same time the quantity y Bx represents 

 more nearly the true area of each strip. 



In the limit, when 8x is made infinitely small, the area of each 

 strip becomes infinitely small, but in order to find the area KLMN 

 an infinitely great number of these strips must be added 

 together. 



Hence A, the area, is the sum of all such terms of the form 

 y $x when $x is made infinitely small, or A = 2t/ $x when 8x 



is infinitely small. But it has already been shown that A = \y dx; 



therefore [ y dx = 2i/ $x when $x is infinitely small. In other 



words, an integral is the limiting value of the sum of an infinitely 

 great number of infinitely small terms. 



110. Tlie Definite Integral. It has been shown that the area 

 under a curve is given by vy dx, and if the law of the curve 



is known that is, y is given as a function of x this integral 

 can be determined and the area expressed as a function of x. 



We have now to consider what we really have when the area 

 is expressed as a function of x, for at present we only know that 

 it represents the area under some part of the curve. Before the 

 value of this area can be found, we have to fix upon its actual 

 position with respect to the axes of reference, and this can be 

 done by fixing upon the initial and final ordinates. 



Thus, if we erect ordinates at x= a and x = b (Fig. 44), we 

 decide upon the actual position of the area and also fix upon the 

 breadth, or the length of the base line. Therefore the values 

 x = a and x = b are values of x which actually decide what the 

 area under the curve will be. 



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