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
Solid of revolution suitable integral. The disk method is used when the slice that was drawn is "perpendicular to" the axis of revolution; i.e. when integrating "parallel to" the axis of revolution. The volume of the solid formed by rotating the area between the curves of and and the lines and about the -axis is given by If (e.g. revolving an area between the curve and the -axis), this reduces to: The method can be visualized by considering a thin horizontal rectangle at between on top and on the bottom, and revolving it about the -axis; it forms a ring (or disc in