perplexus.info :: Calculus : Step Function

Step Function (Posted on 2004-07-07)

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?

Submitted by Brian Smith

Rating: 5.0000 (1 votes)

Solution: (Hide)

Construct f(x) within a size 1 square. The area above each step, is always a square.
The area of these squares can be calculated easily as 1/4, 1/16, 1/64, ..., 1/4^n, etc. Their sum is 1/3.
Since the original square has area 1, the area under the step function is 2/3.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Puzzle Solution	K Sengupta	2023-05-26 02:20:01
Writing tips	Ostin Cram	2020-09-16 02:15:08
edu	Christen Rowland	2020-06-02 05:33:02
re(2): honest effort	sudip	2004-07-09 08:29:54
Solution	Tristan	2004-07-08 15:38:40
re: honest effort	Charlie	2004-07-08 09:33:18
honest effort	sudip	2004-07-08 00:28:51
A similar way:	Gamer	2004-07-07 14:43:08
solution	Charlie	2004-07-07 13:40:06
Answer (Don't Peek)	Richard	2004-07-07 12:23:19

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