(2)
and
 Sn =
1
2
1
[ ( )
(
)
(
)]
(
)
n
n
r
r
h f x
f x
f x
h
f x
=
+
+…+
= ∑ (3)
Here, sn and Sn denote the sum of areas of all lower rectangles and upper rectangles
raised over subintervals [xr–1, xr] for r = 1, 2, 3, …, n, respectively 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