Abstract :
For many years mathematicians have been trying to find a satisfactory way of defining the area of a curved surface. This apparently easy task is full of hidden difficulties; for instance, if one defines area of a surface in a manner analogous to the definition of length of a curved line it turns out that every curved surface has infinite area — which is certainly an unsatisfactory state of affairs. Of the dozen or more definitions of area that have been proposed in the past 50 years, one given by the great mathematician Lebesgue in 1902 has attracted the most attention and has given the most reasonable answers to questions about surface area.
