Page 246 - DMTH401_REAL ANALYSIS

Real Analysis

Notes
Figure 20.1

Obviously, the area enclosed is k (b-a). NOW, suppose that ]a, b[ is further divided into smaller
intervals by inserting points of division, say
a = x, x £ x £ x £ x = b,
£
1
2
3
4
and the function f is defined so as to take a constant value at each of the resulting sub-intervals
i.e.,
Î
ì k , if x [x ,x [
1
1
0
ï k , if x [x ,x [
Î
2
2
1
f(x) = í k , if x [x ,x [
Î
ï 3 2 3
Î
î k , if x [x ,x ]
1
4
4
Further, suppose that d = length of the ith interval ]x , x [ i.e.,
i i i-1
d = x - x , d = x - x , d = x - x , and d = x - x .
1 1 0 2 2 1 3 3 2 4 4 3
Then, we get four rectangular regions and the area of each region is A, = k d , A, = k d , A, = k ,d ,
1 1 2 2 3 3
and A, = k ,d , as shown in Figure 20.2.
4 4
Figure 20.2

240 LOVELY PROFESSIONAL UNIVERSITY