Consider Wallis's method of finding a formula for 1. Let (n/2 sin" x dx. I(n) = %3D From Wallis's Formulas, 3. n – 1\T I(n) n is even (n 2 2) %3D 4 n or 4. - I(n) n is odd (n 2 3). %3D n (a) Find I(n) for n = 2, 3, 4, and 5. What do you observe? %3D (b) Show that I(n + 1) < I(n) for n 2 2. (c) Show that I(2n + 1) lim = 1. 1(2n) (Hint: Use the Squeeze Theorem.) (d) Verify the Wallis Product using the limit in part (C).

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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.