Minimizing arclength with a set of parameters

[TheStealthTarget]

i am trying make a function who's integral is equal to 1 from 0 to 1 and goes threw points (0,0) and (1,0) with the minimal arc length, is tehre anyway to do this?

i have tried a couple things, which iwll get me xome question, i have made a triangle, and i have made many trapazoids, the trapazoids have a small arc length then the triangle, but i am sure there is a curved line that is even smaller, how would you find this curved line.

 

[berkeman]

What arc lengths do you get for the triangle (2 sides), square (3 sides), trapezoid (still 3 sides), whatever (4 sides), etc.? And what do you get for an ellipse? And what do you get for the function A sin(x) appropriately scaled? Do you see any trends that you can use to prove that <whatever> has minimum arc length?

 

[TheStealthTarget]

for a triangle i get an arc length of 2sqrt4.25 = 4.123105626
for a square (or trapazoid with sides that have a slope of 10000000000 or negligible amount) i get 3

i am redoing the trapazoid with a slope of 4, i realized a made a mistake.