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
Solid surface manufactured in sheet form for fabrication into finished countertops, solid surface can also be cast into a variety of shapes, including sinks, shower pans and bathtubs. Sheet goods can also be heated and bent into three-dimensional shapes using a process called thermoforming, which adds to the versatility of the product. Color and design flexibility are key factors when choosing engineered composites over natural stone. In a residential setting, solid surface can be used as kitchen countertops, bathroom vanity tops, and shower and tub surrounds. Countertop fabricators typically join solid surface sheets into desired shapes using a two-part adhesive, after which