A step function f(x) is defined on the interval [0, 1)

From [0, 2/3), the step is 4/5 high.

From [2/3, 8/9), the step is 24/25 high.

From [8/9, 26/27), the step is 124/125 high.

From [1 - 3^-k, 1 - 3^-(k+1)), the step is 1 - 5^-(k+1) high.

What is the exact area under the step function?

The answer to this is similar to the original step function problem. Consider a related function g(x) which is simply 1-f(x), that is:

From [0, 2/3), the step is 1/5 high.
From [2/3, 8/9), the step is 1/25 high.
From [8/9, 26/27), the step is 1/125 high.
From [1 - 3^(-k), 1 - 3^(-(k+1)) ), the step is 5^(-(k+1)) high.

Then the area under g(x) equals (2/3)(1/5)+(2/9)(1/25)+(2/27)(1/125)+... and so this geometric series sums to (2/15)/(1-1/15)=(2/15)/(14/15)=1/7. Thus the area under f(x) is simply 1-1/7 or 6/7.

Posted by Gamer on 2011-08-07 20:24:51