16.7 Integrate

Description

NumericalIntegration.png

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.

To Use Integration Tool
  1. Create a new worksheet with input data.
  2. Highlight the selected data.
  3. Select Analysis: Mathematics: Integrate from the Origin menu to open the Integ1 dialog box.

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.

Dialog Options

Results Log Output Select to output area, peak location, peak width, and peak height to the Results Log.
Recalculate

Controls recalculation of analysis results

  • None
  • Auto
  • Manual

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.

  • Mathematical Area
    The area is the algebraic sum of trapezoids.
  • Absolute Area
    The area is sum of absolute trapezoid values.
Output Quantities

Specify the quantities to be outputted when the Integration Result box is checked (see below).

  • Dataset Identifier
    Choose a dataset identifier for use when the Integration Result box is checked.
  • Beginning Row Index
    Specify whether to output the beginning row index.
  • Ending Row Index
    Specify whether to output the ending row index.
  • Beginning X
    Specify whether to output the beginning X value.
  • Ending X
    Specify whether to output the ending X value.
  • Max Height
    Specify whether to output the maximum height, as computed from the baseline.
  • X at Max Height
    Specify whether to output the X value that corresponds to the maximum height.
  • Area
    Specify whether to output the integration area.
  • FWHM
    Specify whether to output the peak width at half height of the source curve.

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.

  • None
    Do not plot the integral curve.
  • New Graph
    Plot the integral to a new graph.
  • Source Graph
    Plot the integral in the source graph. This option is available when a graph window is the active window.
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.

Algorithm

Numerical integration involves calculating a definite integral by an approximate function:

\int _{a}^{b}f(x)dx

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:

Integ.png

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)]
  • Difference between Mathematical Area and Absolute Area

Given a baseline y=f(x_0 )\!, the mathematical area of f(x)\! 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]

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] |

Integ22.png

As illustrated above, the baseline is y=f(x_0 )\! 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]

From the expression, we determine that A_1\!, A_2\! and A_3\!, which are above the baseline, are positive, but A_4\! and A_5\!, which are beneath the baseline, are negative.

So the mathematical area of this curve should be A_1+A_2+A_3+A_4+A_5\! and the absolute area of this curve should be |A_1|+|A_2|+|A_3|+|A_4|+|A_5|=A_1+A_2+A_3-A_4-A_5\!.