The Riemann integral is a method for calculating the area under a curve defined by a function over a specific interval. It involves dividing the area into small rectangles, calculating their areas, and then summing these areas. As the width of the rectangles approaches zero, the sum converges to the exact area, which is the value of the integral. This concept is named after the mathematician Bernhard Riemann, who formalized this approach in the 19th century. The Riemann integral is foundational in calculus and is used to analyze functions, compute areas, and solve various problems in mathematics and physics.