(a) Use the definition given below with right endpoints to express the area under the curve y = x3 from 0 to 1 as a limit. The area A of the region S that is bounded above by the graph of a continuous function y = f(x), below by the x-axis, and on the sides by the lines x = a and x = b is the limit of the sum of the areas of approximating rectangles. A = lim R, = lim [f(x,)Ax + f(x2)Ax + ... + f(x,)Ax] = lim n- 00 i=1 lim (b) Use the following formula for the sum of cubes of the first n integers to evaluate the limit in part (a). 13 + 23 + 33 + .... + n3 = n(n +

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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.