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September 19th, 2015, 01:46 PM   #1
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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
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September 19th, 2015, 11:42 PM   #2
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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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