Expert: Paul Klarreich - 7/27/2009

Question
Find a continuous, non negative function that has finite area on [0,1] and an infinite arc length on [0,1].

Our teacher gave us a hint saying that it was close to ((sin(1/x))^2 . The reason why this function doesn't work is because there's a hole at 0 but it oscillates infinitely.

Answer
Questioner: Sharon
Country: United States
Category: Advanced Math
Private: No
Subject: Calc 2
Question: Find a continuous, non negative function that has finite area
on [0,1] and an infinite arc length on [0,1].

Our teacher gave us a hint saying that it was close to ((sin(1/x))^2 .
The reason why this function doesn't work is because there's a hole at
0 but it oscillates infinitely
...................................................

Try: y = x (sin(1/x))^2, for  x /= 0, and  y = 0, for x = 0.

Now
1. It does have finite area, obviously, since it is contained in the unit square.

2. It is continuous at  x = 0, since  lim[x->0] (f(x)) = 0

Alas, I do not have a proof that it has infinite arc length.
.............................

Hmm... maybe I do.

Now one cycle of this would have an arc length like

(sin x)^2 from 0 to 2pi, which would go up to 1 and down to 0, and do it four times.  Thus it would have an arc length at least equal to 4.

But its amplitude is equal to x.  So I think you have a sum of arc lengths that will be a harmonic series, which diverges.