Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 557 — #565
14.3. Approximating Sums 557
f.n/
f.n1/
f.3/
f.2/
f.1/
0 1 2 3
n2 n1 n
P
Figure 14.1
n
iD1 f .i/.
The area of the ith rectangle is f .i/. The shaded region has area
For example, 2 x and p x are strictly increasing functions, while maxfx; 2g and
dxe are weakly increasing functions. The functions 1=x and 2 x are strictly decreasing,
while minf1=x; 1=2g and b1=xc are weakly decreasing.
Theorem 14.3.2. Let f W R C ! R C be a weakly increasing function. Define
and
Then
S WWD
I WWD
Z n
Similarly, if f is weakly decreasing, then
1
nX
f .i/ (14.15)
iD1
f .x/ dx:
I C f .1/ S I C f .n/: (14.16)
I C f .n/ S I C f .1/:
Proof. Suppose f W R C ! R C is weakly increasing. The value of the sum S
in (14.15) is the sum of the areas of n unit-width rectangles of heights f .1/; f .2/; : : : ; f .n/.
This area of these rectangles is shown shaded in Figure 14.1.
The value of
I D
Z n
1
f .x/ dx
is the shaded area under the curve of f .x/ from 1 to n shown in Figure 14.2.