Solid angle is much, much farther away. The solid angle subtended by the complete ()-dimensional spherical surface of the unit sphere in -dimensional Euclidean space can be defined in any number of dimensions . One often needs this solid angle factor in calculations with spherical symmetry. It is given by the formula where is the gamma function. When is an integer, the gamma function can be computed explicitly. It follows that This gives the expected results of 4 steradians for the 3D sphere bounded by a surface of area and 2 radians for the 2D circle bounded by a circumference of length
Solid of revolution the curves of and and the lines and about the -axis is given by If (e.g. revolving an area between curve and -axis), this reduces to: The method can be visualized by considering a thin vertical rectangle at with height , and revolving it about the -axis; it forms a cylindrical shell. The lateral surface area of a cylinder is , where is the radius (in this case ), and is the height (in this case ). Summing up all of the surface areas along the interval gives the total volume. When a curve is defined by its parametric form