Dark Fraction of the Moon
The fraction of the Moon that is dark is a function of the phase angle . This function is and can be found by taking an area integral over the circle. A sphere with one hemisphere black, the other hemisphere white, and rotated by is shown in the top-left corner. The area integral being performed is shown in the top-right corner.
For a unit radius sphere, the dividing equatorial line between the light and dark hemispheres projects to when , and to for . Therefore when , the dark area is given by the area integral from the bottom of the sphere, , to the dividing line . Similarly, when the dark area is given by the area integral from the dividing line to the top of the sphere, , and is . Because the total area of the sphere is , dividing these area integrals by gives us the fraction that is dark, . The fraction of the moon that is illuminated is (1 – the fraction that is dark), and is given by .
This quantity is derived in a different way in the following reference. Note that by the double-angle formula, .
 D. B. Taylor, S. A. Bell, J. L. Hilton, and A. T. Sinclair, Computation of the Quantities Describing the Lunar Librations in the Astronomical Almanac, Ft. Belvoir: Defense Technical Information Center, 2010.