Volume 2 : 3 ratio of the volumes of the sphere and cylinder is credited to Archimedes. The volume of a sphere is the integral of an infinite number of infinitesimally small circular disks of thickness "dx". The calculation for the volume of a sphere with center 0 and radius "r" is as follows. The surface area of the circular disk is formula_10. The radius of the circular disks, defined such that the x-axis cuts perpendicularly through them, is or where y or z can be taken to represent the radius of a disk at a particular x value. Using y
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.,