The Method of Mechanical Theorems convenient to use a little bit of coordinate geometry. If a sphere of radius 1 is placed with its center at "x" = 1, the vertical cross sectional radius formula_2 at any "x" between 0 and 2 is given by the following formula: The mass of this cross section, for purposes of balancing on a lever, is proportional to the area: Archimedes then considered rotating the triangular region between "y" = 0 and "y" = "x" and "x" = "2" on the "x"-"y" plane around the "x"-axis, to form a cone. The cross section of this cone is a circle
Volume "dx". The calculation for the volume of a cone of height "h", whose base is centered at (0, 0, 0) with radius "r", is as follows. The radius of each circular disk is "r" if "x" = 0 and 0 if "x" = "h", and varying linearly in between—that is, The surface area of the circular disk is then The volume of the cone can then be calculated as and after extraction of the constants Integrating gives us In differential geometry, a branch of mathematics, a volume form on a differentiable manifold is a differential form of top degree (i.e.,