Error on a integral quantity with noise

Question

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.

Answer 1

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.