GAAAAAAAAAAAAAAH!
Please tell me what is wrong with this integration (the big S represents the integrate sign):
x = ccos(t)
t = arcos(x/c)
dx/dt = -csin(t)
dx = -csin(t)dt
S sqrt(c^2-x^2)dx
= S sqrt(c^2- (ccos(t))^2)dx
= c*S sqrt(1-cos(t)^2)dx
= c*S sin(t)dx
= c*S sin(t)(-csint))dt
= -(c^2)*S sin(t)^2 dt
= -(c^2)(t-sin(t)cos(t))/2
= -(c^2)(arcos(x/c)-sin(arcos(x/c))cos(arcos(x/c)))/2
= -(c^2)(arcos(x/c)-sin(arcos(x/c))x/c)/2
= -(c^2)(arcos(x/c)-sqrt(1-(x/c)^2)x/c)/2
= (-(c^2)arcos(x/c)+cxsqrt(1-(x/c)^2))/2
BUT IT SHOULD BE ARSIN, NOT ARCOS!
Edit: Oh wait damn. When I plugged it into wolfie to double check it's workingness, I used arsin by mistake. This is right. Enjoy your circle integral.