Find the length of the curve:
y=ln(1-x^2), 0<_x<_1/2
s = ∫[0,1/2] ds
= ∫[0,1/2] √(1+y'^2) dx
= ∫[0,1/2] √(1+(2x/(x^2-1))^2) dx
= ∫[0,1/2] (x^2+1)/(x^2-1) dx
now use partial fractions to get
= ∫[0,1/2] 1 + 1/(x-1) - 1/(x+1) dx
= (x + ln|x-1| - ln|x+1|) [0,1/2]
= (1/2 + ln(1/2) - ln(3/2)) - (0+0-0)
= 1/2 - ln3