The integral of a positive function can be interpreted as the area under measure and integration pdf curve. The Lebesgue integral extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined. Long before the advent of the 20th century, mathematicians already understood that for non-negative functions with a smooth enough graph—such as continuous functions on closed bounded intervals—the area under the curve could be defined as the integral, and computed using approximation techniques on the region by polygons.
However, as the need to consider more irregular functions arose—e. Also, one might wish to integrate on spaces more general than the real line. The Lebesgue integral provides the right abstractions needed to do this important job.