Surface area in the study of fractals. Extensions of the notion of area which partially fulfill its function and may be defined even for very badly irregular surfaces are studied in geometric measure theory. A specific example of such an extension is the Minkowski content of the surface. The below given formulas can be used to show that the surface area of a sphere and cylinder of the same radius and height are in the ratio 2 : 3, as follows. Let the radius be "r" and the height be "h" (which is 2"r" for the sphere). formula_7 The discovery of this
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