The unit circle will be parametrized by (cosw, sinw). Provide a point on it, the region cut out by circle, the x-axis, and the line from the origin to this point has covered area w/2.

As you know, the hyperbolic sine and cosine functions are similar to the usual cosine and sine; they follow similar properties. Here's a cool fact that is parallel to the one above.

The parametrization (coshw, sinh w) parametrizes a hyperbola. Provide a point on it, the region cut out by this hyperbola, the x-axis, and the ray to from the origin to that point also has area w/2!