Question

How do you find the length of the curve `y=sqrt(x-x^2)`?

Answer 1

Use algebra to get a length of `pi/2`

Explanation:

`y=sqrt(x-x^2)` is equivalent to

`y^2 = x-x^2` with the restriction `y <= 0`.

This is equivalent to (`y >= 0` on)

`x^2-x+y^2 = 0`.

This is the equation of a circle. Complete the square to get

`x^2-x+1/4+y^2 = 1/4`, or, better yet

`(x-1/2)^2 + y^2 = 1/4`.

With `y >= 0`, this is the upper semicircle centered at `(1/2,0)` with radius `r = 1/2`

The length of the upper semicircle is half the circumference.

`1/2C = 1/2(2pir) = pi*1/2 = pi/2`

Note
I also tried to do this using the integral, but it became too complicated for me to continue when there was a much cleaner solution available.