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The relation between Riemann sum and definite integral is as below.

$$\lim_{n\to \infty} \sum_{k=1}^n f(c_k) \, \Delta x = \int_a^b f(x) \, dx$$

How to determine the interval $(a,b)$ if only the summation is expressed? In other words, how is the interval incorporated in the summation expression?

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  • $\begingroup$ Take the Riemann sum $$\frac{b-a}{n}\sum_{k=0}^n f(a+\frac{k}{n}(b-a))$$ This is a Riemann sum relative to the interval $I=[a,b]$, and the function $f$. But this is also the Riemann sum relative to the interval $J=[0,1]$, and the function $g(x)=(b-a)f(a+x(b-a))$. So the interval is not uniquely determined by the Riemann sum. $\endgroup$ – Kelenner Nov 8 '17 at 14:43
  • $\begingroup$ Then the equation in the OP ceases to be an equation, correct? $\endgroup$ – Vinny Nov 8 '17 at 18:25

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