A partition P = {I0, I1, I2,...,In 1,Tn} of the interval [a, b) is said to be "regular" if the subintervals [2i 1,7] all have the same length given by Az = (b- a)/n. Let P, be a regular partition of (a, b). Show that if a function f is continuous and increasing on (a, b), then U(f, P,) – L(f, P;) = [f(b) – f(a)|Az.

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Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus. 

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.