Solid angle orientation formula_9, and distance from the viewer as: where the surface area of a sphere is . The solid angle of a cone with its apex at the apex of the solid angle, and with apex angle 2, is the area of a spherical cap on a unit sphere For small such that , this reduces to the area of a circle . The above is found by computing the following double integral using the unit surface element in spherical coordinates: This formula can also be derived without the use of calculus. Over 2200 years ago Archimedes proved that the
Gabriel's Horn using Cavalieri's principle before the invention of calculus, but today calculus can be used to calculate the volume and surface area of the horn between and , where . Using integration (see Solid of revolution and Surface of revolution for details), it is possible to find the volume and the surface area : The surface area formula above gives a lower bound for the area as 2 times the natural logarithm of . There is no upper bound for the natural logarithm of as approaches infinity. That means, in this case, that the horn has an infinite surface area. That