Question: How does one calculate the propagated measurement uncertainty to a value that was determined through integration by use of the trapezoidal rule?
My problem: I measure a value $p$ which has some associated absolute uncertainty $\delta p$, i.e., I state the result of my measurement as the best estimate of the quantity $p$ and the range within which I am confident the quantity $p$ lies: $p\pm \delta p$. For this value of $p$ I want to calculate the value $m(p)$, which is calculated as $$\tag{1} m(p)=\int_{p_o}^p \frac{\rho(p)}{\mu(p)} \ dp$$
For physical context, $p$ is fluid pressure, $\rho$ is fluid density, and $\mu$ is fluid viscosity. The values of density and viscosity are pressure dependent as indicated in Eqn(1). For the definite integral in Eqn(1), the integration is from some "base pressure" $p_o$ to the pressure of interest $p$ (where $p>p_o$).
What I have tried: The trapezoidal rule integration formula is $$\tag{2} \int_a^b f(x) \ dx \approx \frac{1}{2} \sum_{i=1}^n (x_i-x_{i-1})(y_i+y_{i-1})$$
Combining (1) and (2) we obtain $$\tag{3} m(p)=\frac{1}{2} \sum_{i=1}^n (p_i-p_{i-1})[(\rho /\mu)_i+(\rho /\mu)_{i-1}]$$
The general equation for the propagation of uncertainty is $$\tag{4} \delta q = \sqrt{\left(\frac{\partial q}{\partial x}\delta x\right)^2+...+\left(\frac{\partial q}{\partial z}\delta z\right)^2}$$ where $x,...,z$ are quantities measured with uncertainties $\delta x,...,\delta z$ and the measured values are used to compute the function $q(x,...,z)$. The uncertainties in $x,...,z$ are independent and random.
As we calculate the partial sums of Eqn(3) we note that the values used for $p_i$ and $p_{i-1}$ have no uncertainty. So our focus is on the uncertainties of calculated values for $(\rho /\mu)$. Denoting the quotient $(\rho /\mu)$ as $R$, the uncertainty in $R$, per Eqn(4), is: $$\tag{5} \delta R = \sqrt{\left(\frac{\partial R}{\partial \rho}\delta \rho \right)^2+\left(\frac{\partial R}{\partial \mu}\delta \mu \right)^2}$$
Thus, we have the uncertainty in the partial sum $S$ computed as $$\tag{6} \delta S =\sqrt{\delta R_i^2 + \delta R_{i-1}^2}$$
Per Eqn(3) we will need to perform this computation for each value of $i$, from $i=1$ to $i=n$, and then compute the summation of all these values. And per Eqn(4) we would calculate the uncertainty in $m(p)$ as $$\tag{7} \delta m(p) = \frac{1}{2} \sqrt{\sum_{i=1}^n (\delta S^2)_i}$$
Not sure if this all makes sense and therefore why I have asked my question: How does one calculate the propagated measurement uncertainty to a value that was determined through integration by use of the trapezoidal rule?
