Assuming the applicability of Fubini's theorem and the multivariate change of variables formula, the disk method may be derived in a straightforward manner by (denoting the solid as D): Cylinder method The cylinder method is used when the slice that was drawn is parallel to the axis of revolution; i.e. when integrating perpendicular to the axis of revolution. The volume of the solid formed by rotating the area between 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. This method may be derived with the same triple integral, this time with a different order of integration: Parametric form When a curve is defined by its parametric form in some interval , the volumes of the solids generated by revolving the curve around the -axis or the -axis are given by Under the same circumstances the areas of the surfaces of the solids generated by revolving the curve around the -axis or the -axis are given by Polar form For a polar curve where , the volumes of the solids generated by revolving the curve around the x-axis or y-axis are The areas of the surfaces of the solids generated by revolving the curve around the -axis or the -axis are given See also Gabriel's Horn Guldinus theorem Pseudosphere Surface of revolution Ungula Notes References () Integral calculus Solids