13.3. Approximating Sums 513
0 1 2 3
^
n� 2 n� 1 n
f.n/
f.n�1/
f.3/
f.2/
f.1/
Figure 13.1P The area of theith rectangle isf.i/. The shaded region has area
n
iD 1 f.i/.
For example, 2 xand
p
xare strictly increasing functions, while maxfx;2gand
dxeare weakly increasing functions. The functions1=xand 2 xare strictly de-
creasing, while minf1=x;1=2gandb1=xcare weakly decreasing.
Theorem 13.3.2.LetfWRC!RCbe a weakly increasing function. Define
SWWD
Xn
iD 1
f.i/ (13.15)
and
IWWD
Zn
1
f.x/dx:
Then
ICf.1/SICf.n/: (13.16)
Similarly, iffis weakly decreasing, then
ICf.n/SICf.1/:
Proof. Supposef WRC !RCis weakly increasing. The value of the sumS
in (13.15) is the sum of the areas ofnunit-width rectangles of heightsf.1/;f.2/;:::;f.n/.
This area of these rectangles is shown shaded in Figure 13.1.
The value of
ID
Zn
1
f.x/dx
is the shaded area under the curve off.x/from 1 tonshown in Figure 13.2.