# Error on a integral quantity with noise

First of all sorry if this is the wrong place to ask this question, I went to a few stack sites and thought here it would be more suitable.

My problem:

I have a physical quantity $F$ that depends on $x$. At each $x$ I have an error $dF$ in the measurement of $F$. The quantity I want to measure is defined as an integral:

$$M = \int\limits_{x_1}^{x_2} F(x)dx$$

I would like to be able to compute the error $dM$ in $M$.

I was thinking about estimating the same integral replacing $F(x)$ by $F(x)+dF(x)$ for the upper error and $F(x)-dF(x)$ for the lower error but I would like to know if there is a better way to make this error estimation.

• I do not have a definite answer for your question, but I definitely think this is the appropriate stackexchange for this question. – BlaB Apr 17 '18 at 13:51
• The key question is what you know about your noise. Can it always be positive? Does it oscillate? Do you have any information about its statistics? If you do, then that's the key starting point for estimating the error $dM$. – Wolfgang Bangerth Apr 18 '18 at 6:59

Assuming that you mean the following inequality in your prompt $$|F_\mathrm{true}(x) - F(x)| \le |dF(x)| \qquad \forall x,$$ a simple bound for $dM$ is the following $$dM \equiv | M_\mathrm{true} - M | = \left|\int_{x_1}^{x_2} F_\mathrm{true}(x) - F(x) dx\right| \le \int_{x_1}^{x_2} |dF(x)| dx.$$ This also assumes that $dF$ is absolute integrable.