A step function f(x) is defined on the interval [0, 1)
From [0, 1/2), the step is 1/2 high.
From [1/2, 3/4), the step is 3/4 high
From [3/4, 7/8), the step is 7/8 high
From [ 1  1/(2^k), 1  1/(2^(k+1)) ) the step is 1  1/(2^(k+1)) high
What is the exact area under the step function?
The width of each vertical strip making up the area is 1/2^(k+1), while the height is 11/2^(k+1). The area of each strip is therefore 1/2^(k+1)  1/4^(k+1), with k going from zero to infinity.
Let a be the sum of all the positive terms and b be the sum of the absolute values of the negative terms. The answer will be a  b.
a = 1/2 + 1/4 + 1/8 + 1/16 + ...
2a = 1 + a
a = 1
and
b = 1/4 + 1/16 + 1/64 + 1/256 + ...
4b = 1 + b
3b = 1
b = 1/3
Total area = a  b = 1  1/3 = 2/3.

Posted by Charlie
on 20040707 13:40:06 