194 PRACTICAL MATHEMATICS 



Taking the relation Area = \y dx, and after the integration 



has been performed, x is given the value a, the result will give 

 the value of the area HOLK, the area which is bounded by the 

 ordinates at x = and x = a. 



Also taking the same relation Area = I y dx, and after the 



integration has been performed, x is given the value b, the result 

 will give the value of the area HOMN, the area which is bounded 

 by the ordinates at x = and x = b. 



Now area KLMN = area HOMN - area HOLK 



= \y dx (when x = b) \y dx (when x = a) 

 or area KLMN = I y dx, where, after the integration has been 



Jo, 



performed, x is given the values b and a respectively and the 

 difference of the two results is taken. 



We notice now that integration can be performed with respect 

 to x over a definite range, or between two definite values of x. 

 These two limiting values of x are spoken of as the superior and 

 inferior limits respectively, and the result obtained by replacing 

 x by the value of the inferior limit must be subtracted from the 

 result obtained by replacing x by the value of the superior limit. 

 It must be clearly understood, however, that before we can sub- 

 stitute the values of the limits we must have performed the 

 necessary integration. 



111. Areas. Let P and Q be two points on a curve (Fig. 45), 

 the co-ordinates of P being (a, h) and the co-ordinates of Q being 

 (b, k). Let A be the area bounded by the curve, the axis of x, 

 and the ordinates h and k. Let B be the area bounded by the 

 curve, the axis of y, and the abscissae a and b. Then A can be taken 

 as the sum of vertical strips of area y $x taken from x = a to 

 x = b, while B is the sum of horizontal strips of area x y taken 

 from y = h to y = k, 



and A 



= \ydx 



while B = 

 A study of the figure will show that 



b 



x dy + ah = bk 



[ y dx + [ 



Ja JJt 



