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Uncertainty of an Integral


The uncertainty of an integral that is based on measured data with inherent uncertainty can be estimated using the Uncertainty Propagation Table command. It is necessary to include all calculated variables in a Parametric Table, including the variable(s) involved in the integrand, the integration variable, and the independent variable.  Values of the integration variable must be entered into the Parametric table.  Uncertainties can be assigned to any of the variables, as indicated in the Uncertainty Propagation Table command.  EES will calculate the contribution to the uncertainty in the integration variable for row in the table.  The total uncertainty associated with the integral is approximated as the square root of the sum of the squares of the uncertainties for each row.  EES displays the total uncertainty calculated in this way in the Sum row of the table.  Use the $SUMROW ON directive to display the sum row in the Parametric table.


As a simple example, we will determine the uncertainty in W where



where k and x are measured at 11 points.  x is precisely measured, but the measured values of k (which has a nominal value of 10 N/m) each have uncertainty.


Enter the integral equation into the Equations window.




Create a Parametric table with columns for k, W, and x.  Enter the values of k and x into the table as shown.



Select Uncertainty Propagation Table from the Calculate menu and select k as the measured variable.



Click the Set uncertainties button and enter the uncertainty for each value of k.



Click OK on both dialogs. The uncertainty calculations will proceed and the Parametric table will be filled out as shown.



The uncertainty value for W for each row is the uncertainty associated with the contribution of the integral.  The total uncertainty is the square root of the sum of the squares of the uncertainty values for all rows.  This value appears in the sum row, which is 0.000883 N/m in this example.


Note that the result obtained in this way assumes that each value of k is independent and each value has a separate uncertainty.  The values of k vary as the integral proceeds.  The result that would be obtained if k were a constant with an uncertainty of 0.05 N/m can be determined analytically as the product of the derivative of W with respect to k and the uncertainty of k.  In this case, that result is 0.045*0.05 = 0.0225 N/m.  The reason it is larger is that only one k value is used rather than 11 as in the example.


The integral could also be computed with uncertainty assigned to the integration variable, x.