4. Express each of the following limits as a definite integral int_0^1 f(x) dx and use it to evaluate the limit. (a) lim_{n o infty} frac{1}{n} left{ sin(frac{pi}{n}) + sin(frac{2pi}{n}) + cdots + sin(frac{kpi}{n}) + cdots + sin(frac{(n-1)pi}{n}) + sin(frac{npi}{n}) ight} (b) lim_{n o infty} left{ frac{1}{n+1} + frac{1}{n+2} + cdots + frac{1}{n+k} + cdots + frac{1}{n+n} ight} (c) lim_{n o infty} frac{1^2 + 2^2 + cdots + n^2}{n^3} formula to use int_a^b f(x) dx = lim_{n o infty} S_n = lim_{n o infty} sum_{i=1}^n f(x_i^*) Delta x. Answers: 4. (a) frac{2}{pi} (b) ln 2 (c) frac{1}{3}.
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(a) The limit expression is not clear, so it's impossible to express it as a definite integral or evaluate the limit. Show more…
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