Integration tool performs numerical integration on the active data plot using the trapezoidal rule. You can choose to calculate the Mathematical Area (the algebraic sum of trapezoids) or an Absolute Area (the sum of absolute trapezoid values). Missing values are ignored.
The X-Function Integ1 is called to perform the calculation. The user has the option to specify that the area, peak location, peak width, and peak height (maximum deflection from the X axis), are written to the Result Log. In addition, you can choose to integrate using a simple baseline defined by a straight line connecting the end points of the curve and to create a plot of the integral curve.
 | Note: This tool uses pure, math-based integration which may produce unexpectedly negative results for area when the X values used in the integration are in descending order. This is normal behavior due to the nature of the integration calculations. |
| Results Log Output | Select to output area, peak location, peak width, and peak height to the Results Log. |
|---|---|
| Recalculate |
Controls recalculation of analysis results
For more information, see: Recalculating Analysis Results |
| Input |
Specify the input data to be integrated. For help with range controls, see: Specifying Your Input Data |
| Use End Points Straight Line as Baseline |
Specify whether to create a straight line that crosses the end points and use it as the baseline for the integration. |
| Area Type |
Specify the integral area type. Please see the Algorithm section below, for more details.
|
| Output Quantities |
Specify the quantities to be outputted when the Integration Result box is checked (see below).
Note: The beginning and ending X values and the integrated area are output to the Comments row of the integration results column, regardless of whether the Output Quantities Beginning X, Ending X and Area are enabled. |
| Integral Curve Data |
Specify the range to output the cumulative result. |
| Integration Result |
Specify whether output the intergration result in a report sheet. |
| Plot Integral Curve |
Specify whether to plot the integral curve, and where to plot the integral curve.
|
| Rescale Source Graph |
Rescale the source graph when the integral is plotted into it. This check-box is available when the Plot Integral Curve is Source Graph. |
Numerical integration involves calculating a definite integral by an approximate function:
Since the original data are discrete, we use a pair of adjacent values to form a trapezoid for approximating the area beneath the segment of the curve defined by the two points:
As illustrated above, the curve is divided into pieces and we calculate the sum of each trapezoid to estimate the integral by:
![\int _{x_1}^{x_n}f(x)dx \approx \sum _{i=1}^{n-1}( x_{i+1} -x_i) \frac{1}{2}[f(x_{i+1})+f(x_i)]](/origin-help/en/images/Integrate/math-02e1de68d6e5a54203ad7578c5a02844.png?v=0 "\int _{x_1}^{x_n}f(x)dx \approx \sum _{i=1}^{n-1}( x_{i+1} -x_i) \frac{1}{2}[f(x_{i+1})+f(x_i)]")
Given a baseline 
, the mathematical area of 
can be calculated by
![\int _{x_1}^{x_n} \left[f \left( x \right)-f \left( x_0 \right) \right] \,dx \approx \sum _{i=1}^{n-1} \frac{1}{2} \left( x_{i+1} -x_i \right) \left[ \left( f \left( x_{i+1} \right) -f \left( x_0 \right) \right) + \left( f \left( x_i \right) -f \left( x_0 \right) \right) \right]](/origin-help/en/images/Integrate/math-a25d330f4319359dc2e20213e8096e19.png?v=0 "\int _{x_1}^{x_n} \left[f \left( x \right)-f \left( x_0 \right) \right] \,dx \approx \sum _{i=1}^{n-1} \frac{1}{2} \left( x_{i+1} -x_i \right) \left[ \left( f \left( x_{i+1} \right) -f \left( x_0 \right) \right) + \left( f \left( x_i \right) -f \left( x_0 \right) \right) \right]")
If the sum of each trapezoid's area absolute value is computed, we can get the absolute area:
![\int _{x_1}^{x_n} | \left[f \left( x \right)-f \left( x_0 \right) \right] | \,dx \approx \sum _{i=1}^{n-1} \frac{1}{2} \left( x_{i+1} -x_i \right) | \left[ \left( f \left( x_{i+1} \right) -f \left( x_0 \right) \right) + \left( f \left( x_i \right) -f \left( x_0 \right) \right) \right] |](/origin-help/en/images/Integrate/math-2c032ecbce6257d0ec1dfd6f4510b0ad.png?v=0 "\int _{x_1}^{x_n} | \left[f \left( x \right)-f \left( x_0 \right) \right] | \,dx \approx \sum _{i=1}^{n-1} \frac{1}{2} \left( x_{i+1} -x_i \right) | \left[ \left( f \left( x_{i+1} \right) -f \left( x_0 \right) \right) + \left( f \left( x_i \right) -f \left( x_0 \right) \right) \right] |")
As illustrated above, the baseline is and the curve is divided into five trapezoids (or triangles). The area of each trapezoid (or triangle) is calculated by
![A_i = \frac{1}{2} \left( x_{i+1} -x_i \right) \left[ \left( f \left( x_{i+1} \right) -f \left( x_0 \right) \right) + \left( f \left( x_i \right) -f \left( x_0 \right) \right) \right]](/origin-help/en/images/Integrate/math-741a03a31a9c340fc809dbde0bfd9d03.png?v=0 "A_i = \frac{1}{2} \left( x_{i+1} -x_i \right) \left[ \left( f \left( x_{i+1} \right) -f \left( x_0 \right) \right) + \left( f \left( x_i \right) -f \left( x_0 \right) \right) \right]")
From the expression, we determine that 
, 
and 
, which are above the baseline, are positive, but 
and 
, which are beneath the baseline, are negative.
So the mathematical area of this curve should be 
and the absolute area of this curve should be 
.