The total surface area of a solid is the sum of the areas of all of the faces or surfaces that enclose the solid. The faces include the tops and bottoms (bases) and the remaining surfaces. The lateral surface area of a solid is the surface area of the solid without the bases.
A solid object's surface area is a measurement of the overall space that the object's surface takes up. In comparison to the definition of arc length for one-dimensional curves or the definition of surface area for polyhedra (i.e., objects with flat polygonal faces), where the surface area is equal to the sum of the areas of its faces, the mathematical definition of surface area in the presence of curved surfaces is significantly more complex. The surface area of smooth surfaces, such spheres, is determined using their representation as parametric surfaces. Partial derivatives and double integration are used in this definition of surface area, which is based on infinitesimal calculus techniques.
Henri Lebesgue and Hermann Minkowski sought a broad notion of surface area.
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