16.8 2D Volume Integrate (Pro Only)

Overview

2D Volume Integration tool calculates the volume between the plane Z=0 and the matrix surface, by two-dimensional integration. Two-dimensional volume integration can be performed on a matrix window or on a plot of matrix data.

To Use 2D Volume Integration Tool

1. Create a new matrix with data.
2. Activate the matrix.
3. Select Analysis: Mathematics: 2D Volume Intergrate from the Origin menu to open the integ2 dialog.
4. Choose your options and click OK. The X-Function integ2 is called to perform the calculation.

Dialog Options

Input Matrix	The operating matrix.
Interpolate for Missing Value	Trim missing values if this parameter is true.

Algorithm

This function computes the volume beneath the matrix surface using a numeric integral method.

For a continuous surface \(z=f(x,y),(x,y)\in \sigma\), the volume beneath it can be computed as:

\(\iint_{(\sigma)}f(x,y)dxdy\),

Using a numeric method, it can be written as:

\[\iint_{(\sigma)}f(x,y)dxdy=\lim_{\Delta x \to 0}\lim_{\Delta y \to 0} \sum_{i=0}^{m-1} \sum_{j=0}^{n-1} f(x_{i,}y_j)\Delta x\Delta y\approx \sum_{i=0}^{m-1} \sum_{j=0}^{n-1} f(x_i,y_j)\Delta x\Delta y\]

where the M, N is the number of the rows and columns of the matrix respectively. In the actual process of computing,

\[\left( f\left( x_i,y_j\right) +f(x_i,y_{j+1})+f(x_{i+1},y_j)+f(x_{i+1},y_{j+1})\right)/ 4 \]

is used instead of \(f(x_i,y_j)\!\).