My Math Forum Integral - Euler Maclaurin formula

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 September 19th, 2015, 01:46 PM #1 Newbie   Joined: Sep 2015 From: USA Posts: 1 Thanks: 0 Integral - Euler Maclaurin formula greetings, I was computing a definite integral using the fundamental theorem, and, out of curiosity, I attempted to evaluate the integral using the definition of the definite integral. when evaluating this integral using the fundamental theorem, the answer is -32/15 . here is the integral. $\displaystyle$$\int_{4}^{0} \sqrt{t} (t-2) dt$ I could not compute this definite integral using the definition of the definite integral, as it involves rewriting $\displaystyle \sum_{i=0}^{n} \sqrt{i}$ before taking the limit. wolfram alpha defines this sum as the generalized harmonic number. it was suggested to me that the euler-maclaurin formula for integrals could evaluate this specific definite integral. I was wondering if someone could assist in computing this definite integral using the formula. I arrived at a rather rough approximation, 2.8312, and as I do not see this formula applied using specific examples, I assume my calculations are incorrect. https://en.wikipedia.org/wiki/Euler%...laurin_formula
 September 19th, 2015, 11:42 PM #2 Global Moderator   Joined: Dec 2006 Posts: 20,835 Thanks: 2162 Note that the integrand is negative for 0 < $x$ < 2. Can you post your attempted evaluation based on the definition of the indefinite integral, so that we can identify any mistakes?

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