Integration

See Also: Regression Interpolation Differentiation Data table

The “Integration” dialog box is shown below. These data are from Example 3 - "Heat capacity" in the Polymath REG Regression Program. To find the integral of a dependent variable over a range of an independent variable you should select first the independent variable’s name and the dependent variable’s name. After specifying a lower and upper limit for the independent variable the integral (area under the curve) is calculated over the specified region. Note that both the lower and the upper limits of the independent variable value must lie inside the region where data is available.

Three algorithms are available for integration: Gauss-Legendre orthogonal polynomial approximation (GAULEG), Simpson’s rule (QSIMP and SIMPSON) and Cubic spline approximation (SPLINE). The various methods are discussed in detail in pp. 123-145 of Press et al. For data that can be represented with a smooth function with fairly regular shape the Cubic spline approximation can be used. The program gives a warning message if there is large differences in the values obtained using the different techniques.

In the screen display shown above, the integral CpdT is calculated from the lower bound of 120 K up to the upper bound of 180 K for the data of Example Problem 3. The calculated integral value (using a cubic spline function) is 680.37 and the estimated error is essentially zero. Calculating the integral using alternative methods shows that the calculated value is very accurate indeed.