Surface of revolution never negative, the area is given by If the curve is described by the function , , then the integral becomes for revolution around the -axis, and for revolution around the "y"-axis (using ). These come from the above formula. For example, the spherical surface with unit radius is generated by the curve , , when ranges over . Its area is therefore For the case of the spherical curve with radius , rotated about the -axis A minimal surface of revolution is the surface of revolution of the curve between two given points which minimizes surface area. A basic
Surface area but not on its position and orientation in space. This means that surface area is invariant under the group of Euclidean motions. These properties uniquely characterize surface area for a wide class of geometric surfaces called "piecewise smooth". Such surfaces consist of finitely many pieces that can be represented in the parametric form with a continuously differentiable function formula_4 The area of an individual piece is defined by the formula Thus the area of "S" is obtained by integrating the length of the normal vector formula_6 to the surface over the appropriate region "D" in the parametric "uv" plane. The