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} (t2) 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 eulermaclaurin 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?


Tags 
euler, formula, integral, maclaurin 
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