Volume as the disk radius, the volume of the sphere can be calculated as Now Combining yields formula_15 This formula can be derived more quickly using the formula for the sphere's surface area, which is formula_16. The volume of the sphere consists of layers of infinitesimally thin spherical shells, and the sphere volume is equal to The cone is a type of pyramidal shape. The fundamental equation for pyramids, one-third times base times altitude, applies to cones as well. However, using calculus, the volume of a cone is the integral of an infinite number of infinitesimally thin circular disks of thickness
Volume element compute the area of a set "B" lying on the surface by computing the integral Here we will find the volume element on the surface that defines area in the usual sense. The Jacobian matrix of the mapping is with index "i" running from 1 to "n", and "j" running from 1 to 2. The Euclidean metric in the "n"-dimensional space induces a metric formula_34 on the set "U", with matrix elements The determinant of the metric is given by For a regular surface, this determinant is non-vanishing; equivalently, the Jacobian matrix has rank 2. Now consider a change of