With a programmable calculator (or a computer), it is possible to evaluate the e

Question

With a programmable calculator (or a computer), it is possible to evaluate the expressions for the sums of areas of approximating rectangles, even for large values of n, using looping. (On a TI use the Is> command or a For-EndFor loop, on a Casio use Isz, on an HP or in BASIC use a FOR-NEXT loop.) Compute the sum of the areas of approximating rectangles using equal subintervals and right end points for n = 10, 30, 50, and 100. (Round your answers to four decimal places.)

Explanation / Answer

For some n, the interval (width) of each rectangle = pi/(2*n)

The height of the rectangle (taking the right end) is 2*cos(k*pi/(2*n))

Hence the sum of all rectangular ares will be (pi/(2*n))*2*cos(1*pi/(2*n)) + (pi/(2*n))*2*cos(2*pi/(2*n)) + .... (pi/(2*n))*2*cos(n*pi/(2*n))

= (pi/n) [ cos(pi/2n) + cos(2pi/2n) + ... cos(n*pi/2n) ]

Now for n = 10, we get 1.83881

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