Question

If f(0) is 0 and f(2) is 0 and the integral of f(x) from 0 to 2 is 2, what is the smallest arc length possible?

Answer 1

What's the length of the shortest path between two points?

Answer 2

Am confused. I kind of need to find what the function is.

Answer 3

@SithsAndGiggles I believe you missed the part where it says integral from 0 to 2 is 2. So certainly it is not going to be a straight line

Answer 4

Oh hold up, I see what you mean...

Answer 5

Must have read that as \(\sqrt{1+[f'(x)]^2}\) for whatever reason

Answer 6

i was thinking of the sin curve because using sin(pi*x), i will get 0 for both 0 and 2.which solves my f(0)=0 and f(2) =0 However if the function is integrated, i need to get an area of 2 and i dont know how i can manipulate the function to get an area of 2.

Answer 7

I was thinking this curve:

We want to minimize the area in some sense to get the shortest length

Answer 8

@sourwing yes was just about to suggest a triangular path

Answer 9

Doesn't have to be a triangular shape. As long as the area is 2, we can change the shape a little bit to make it an arc while still retaining the shortest length

Answer 10

I mean certainly, we don't want this curve:

Answer 11

I figured a triangular contour would be the logical step from a straight line - not to say that this is completely right. Maybe a semi-elliptical path could work too.

Answer 12

Trapezoidal perhaps?

Answer 13

Curious to see how this can be set up as an optimization problem...

Answer 14

Can you guys take a quick look at this website. My problem is very similar except for my limits of integration. http://ecademy.agnesscott.edu/~lriddle/arc/contest.htm

Answer 15

Well there you go, a partial circle.

Answer 16

interesting problem though :)

Answer 17

Am really stuck. I have to come up with the function and all the ones i've tried dont seem to work

Answer 18

Part of a circle sounds logical, because it has a constant radius of curvature.
Any where there is a change of curvature, we could get the same area for a shorter arc.
The formal proof would be another challenge.