Spiral
Each point (Y) in the spiral is found by taking the right triangle whose sides are the previous point (X), the origin (A), and a new point found by intersecting a unit circle rooted at (X) with the line perpendicular to (AX). Thus the quadrature (length squared) of each successive hypotenuse increases by 1.
Since the height of each triangle is 1, the area is equal to half its base, and the area covered by the whole spiral is the sum of the square roots of the natural numbers, divided by two.... until you get to point (S), when the triangles start to overlap.
Is there a function that describes the area covered by the entire spiral up to point (X)?