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Analysis: Riemann Sums

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(a) Given a bounded function fâ?¶[a,b]â?'R, and a partition
Pâ?¶a=?_0<?_1<?_2<â?¯<?_n=b,
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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The expert examines Riemann sums.

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