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    Analysis: Riemann Sums

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    (a) Given a bounded function fâ?¶[a,b]â?'R, and a partition
    Define the lower Riemann sum L(f,P) and the upper Riemann sum U(f,P) of f with respect to P, as well as the lower and upper integral of f. When is f called Riemann integrable and how is â?«^b_a f(x)dx defined in this case?

    (b) Show that g(x)=tan?(x) is a primitive for f(x)=1/cos^2 (x).

    (c) Determine
    S=lim(nâ?'â??)? 1/n (1+1/(cos^2 (Ï?/4n))+1/(cos^2 (2 Ï?/4n))+â?¯+1/(cos^2 ((n-1)Ï?/4n))).
    Justify your answer.

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    Solution Summary

    The expert examines Riemann sums.