Riemann integrability is a mathematical concept that describes the ability to calculate the area under a curve using a specific method called the Riemann integral. It is used to find the exact value of an integral, which is the limit of a sum of infinitely many small rectangles under a curve.
Riemann integrability differs from other types of integration, such as Lebesgue integration, in the way it approximates the area under a curve. While Lebesgue integration partitions the domain into smaller subsets, Riemann integration divides the domain into equal intervals and uses the value of the function at the endpoints of each interval to calculate the area.
A function must be continuous on a closed interval and bounded in order to be Riemann integrable. This means that the function must have a defined value at every point within the interval and its values must not exceed a certain limit.
Riemann integrability is important in mathematics because it provides a way to calculate the exact value of an integral, which is a fundamental concept in calculus and other areas of mathematics. It allows for precise calculations and can be applied to a variety of functions and problems.
Riemann integrability is a concept that describes the ability to calculate the area under a curve using a specific method, while Riemann sum is a numerical technique used to approximate the area under a curve using a finite number of rectangles. Riemann integrability provides the exact value of an integral, while Riemann sum provides an estimate.