In view of the inequality (1) for an arbitrary subinterval [xr–1, xr], we have
sn < area of the region PRSQP < Sn (4)
As n → ∞ strips become narrower and narrower, it is assumed that the limiting
values of (2) and (3) are the same in both cases and the common limiting value is the
required area under the curve Symbolically, we write
limSn
n→∞
 = lim
n
n
s
→∞
 = area of the region PRSQP = 
( )
b
∫a f x dx (5)
It follows that this area is also the limiting value of any area which is between that
of the rectangles below the curve and that of the rectangles above the curve