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Consider a quantity $f(x)$ that has an absolute error $\delta x(x)$. The parentheses indicates that $\delta x$ varies with $x$. What is the uncertainty in $\int_0^x f(x) dx$?

For simplicity let's assume $x$ is known accurately. I found formulae about errors summing two quantities A and B, but I didn't find any reference when A and B were infinitesimals as would be in the case of an integral.

What am I trying to do: I have a quantity in a table with columns "$x$", "$f(x)$" and "$\delta(x)$". I am trying to compute the error in $\int_0^x f(x) dx$

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The integral is just a fancy way of expressing the sum $\sum f(x_i) \cdot \Delta x_i$, where we commonly assume that the intervals $\Delta x_i$ are equivalent-distantly spaced. Therefore, a simple and robust method is to use the sum to estimate the error. It might not look fancy, but it certainly is a straight forward method to estimate the error.

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  • $\begingroup$ Wouldn't it be better to use a sum of squared uncertainties? (If uncertainty in this case is proportional to standard deviation) $\endgroup$
    – Redirectk
    Commented Jan 28, 2023 at 17:03
  • $\begingroup$ Why do you believe that the squared uncertainties are a better approx. to the integral given in the question? $\endgroup$
    – Semoi
    Commented Jan 28, 2023 at 21:52
  • $\begingroup$ I should clarify that my comment is expressing a doubt in my knowledge and perhaps you will be able to point out any lack in my reasoning. I thought that if an uncertainty is taken to be the standard deviation of a measurement (may not be the case here) then the uncertainties should be summed in variance space. stats.stackexchange.com/questions/25848/… $\endgroup$
    – Redirectk
    Commented Jan 28, 2023 at 22:22
  • $\begingroup$ The link, which you provided stats.stackexchange.com/questions/25848/… applies to the sum of two independent random variables, which are normally distributed. However, in particular the independence is questionable in the question we are discussing here: Thus, I strongly recommend taking a conservative approach and to calculate an upper bound of the uncertainty. This is obtained by adding the absolute errors. $\endgroup$
    – Semoi
    Commented Jan 31, 2023 at 11:14

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