Decomposing the arc length of a circular arc segment

[jumbo1985]

A particle travels along a circular arc segment centered at the origin of the Cartesian plane with radius R, a start angle θ₁ and an end angle θ₂ (with θ₂ ≥ θ₁ and Δθ = θ₂ - θ₂ ≤ 2π). The total distance traveled is equal to the arc length of the segment: L = R(Δθ).

I would like to find the distance covered by the particle along the X axis and the distance covered by the particle along the Y axis.

I'm not sure how to do this unless I break up the arc at each quadrant crossing and analyze the pieces separately.

Any tips are greatly appreciated.

 

[mfb]

Distance covered as in "if you go back and forth you count it twice"? You can write down an integral that works in general, but analyzing 2 special cases is easier.

 

[jumbo1985]

Yes, exactly - If you go back and forth you count it twice. I'm looking for one elegant expression in terms of x and in terms of y but that may not be possible?