In this problem you will calculate | 3x dx by using the formal definition of the definite integral: n f(x) dx lim E f(x)A. %3D n-00 k=1 (a) The interval [0, 3] is divided into n equal subintervals of length Ax. What is Ax (in terms of n)? Ax = (b) The right-hand endpoint of the kth subinterval is denoted x. What is x (in terms of k and n)? (c) Using these choices for x and Ax, the definition tells us that 3 3x dx = lim E f(x;)Ax n-00 k=1 What is f(x)Ax (in terms of k and n)? f(xt)Ax = (d) Express f(x)Ax in closed form. (Your answer will be in terms of n.) k=1 n E f(xt)Ax = k=1 (e) Finally, complete the problem by taking the limit as n → o of the expression that you found in the previous part. 3 3x dx = lim E f(x;)Ax %3D n-00 k=1

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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.