109. The Meaning of an Integral. 



FIG. 44, 



Let P, Q be two points taken on the curve */=/(#), Fig. 44, 

 and let the co-ordinates of P be x, y, and of Q x + dx, y + dy. 

 When the ordinates are drawn to P and Q, the strip PQTS is 

 produced, the top part of this strip being bounded by the arc PQ 

 of the curve. If the breadth &r is small, the strip may be 

 approximately taken as a trapezium. 



Area of the strip 



If &r is taken as being very small, then 



and can be neglected in comparison with y. 



Hence the area of the strip or SA = y So?. 



is a l so verv small 



Then 

 In the limit when 



is made infinitely small 

 dA 



Therefore A, the area under the curve, is a function of x, which, 

 when differentiated with respect to x, will give y. Now as integra- 



192 



