Problem

You need to numerically integrate a double integral (for example, to compute the volume of some arbitrary shape or the volume under a surface).

Solution

Separate the problem into two parts, each involving only single integrals, and then apply the numerical integration techniques discussed earlier in this chapter.

Discussion

Computing multiple integrals numerically can be challenging, especially when triple or higher integrals and involved. To the best of my knowledge, there are no general-purpose numerical integration formulas that handle multiple integrals. There is a method called *Monte Carlo* integration that involves taking a lot of random samples within the volume or domain being integrated and using the distribution of those samples to estimate the volume or whatever is of interest in higher dimensions. Alternatively, you can break up a multiple integral into successive single integrals and apply the standard numerical integration techniques discussed in this chapter.

To demonstrate this latter approach, consider the surface shown in Figure 10-3.

Figure 10-3. 3D surface

Let's assume you want to compute the volume under that surface bounded by y = [0,1] and x = [0,1]. This is a three-dimensional problem, but instead of computing a multiple integral, you can compute a series of single integrals.

The first step is to consider cross-sections of this surface at successive y-values. Each cross-section, at each y-value, will yield a curve. We can compute the area under each of these curves using a standard integration rule. You'll end up with a new set of data that represents cross-sectional areas as a function of y. You can then integrate the resulting curve of areas to get the volume under the surface.

Figure 10-4 shows the data that represents the surface in Figure 10-3.

The table contains z-values for corresponding x- and y-values. As outlined a moment ago, the first task is to compute cross-sectional areas corresponding to cross-sections

Figure 10-4. Data for 3D surface

taken of the surface at each y-value. This means that for each y-value, held constant, we can integrate along the x-axis to compute the area under the cross-section curve.

The row labeled `Areas` on row 16 of the spreadsheet shows the results of integrating the cross-section curves for each y-value. I applied the trapezoidal rule here, using formulas like `=$D$1*SUMPRODUCT($A$5:$A$15,D5:D15)`. Cells D16 through N16 contain formulas just like this one to compute each cross-sectional area, that is, at each y-value.

If you plot the resulting cross-sectional areas versus the y-values, you end up with a curve like that shown in Figure 10-5.

This plot was generated using the y-data in the cell range D4:N4 and the area data in the cell range D16:N16. The area under this curve is the volume under the surface. So now all you need to do is integrate this cross-sectional area curve to get the volume. Cell D19 contains the formula `=D2*SUMPRODUCT(D17:N17,D16:N16)`, which performs the integration of the cross-sectional area curve, yielding the volume under the surface.

That's all there is to it. This technique is fairly straightforward and readily implemented in Excel.

Figure 10-5. Cross-sectional area curve

Using Excel

- Introduction
- Navigating the Interface
- Entering Data
- Setting Cell Data Types
- Selecting More Than a Single Cell
- Entering Formulas
- Exploring the R1C1 Cell Reference Style
- Referring to More Than a Single Cell
- Understanding Operator Precedence
- Using Exponents in Formulas
- Exploring Functions
- Formatting Your Spreadsheets
- Defining Custom Format Styles
- Leveraging Copy, Cut, Paste, and Paste Special
- Using Cell Names (Like Programming Variables)
- Validating Data
- Taking Advantage of Macros
- Adding Comments and Equation Notes
- Getting Help

Getting Acquainted with Visual Basic for Applications

- Introduction
- Navigating the VBA Editor
- Writing Functions and Subroutines
- Working with Data Types
- Defining Variables
- Defining Constants
- Using Arrays
- Commenting Code
- Spanning Long Statements over Multiple Lines
- Using Conditional Statements
- Using Loops
- Debugging VBA Code
- Exploring VBAs Built-in Functions
- Exploring Excel Objects
- Creating Your Own Objects in VBA
- VBA Help

Collecting and Cleaning Up Data

- Introduction
- Importing Data from Text Files
- Importing Data from Delimited Text Files
- Importing Data Using Drag-and-Drop
- Importing Data from Access Databases
- Importing Data from Web Pages
- Parsing Data
- Removing Weird Characters from Imported Text
- Converting Units
- Sorting Data
- Filtering Data
- Looking Up Values in Tables
- Retrieving Data from XML Files

Charting

- Introduction
- Creating Simple Charts
- Exploring Chart Styles
- Formatting Charts
- Customizing Chart Axes
- Setting Log or Semilog Scales
- Using Multiple Axes
- Changing the Type of an Existing Chart
- Combining Chart Types
- Building 3D Surface Plots
- Preparing Contour Plots
- Annotating Charts
- Saving Custom Chart Types
- Copying Charts to Word
- Recipe 4-14. Displaying Error Bars

Statistical Analysis

- Introduction
- Computing Summary Statistics
- Plotting Frequency Distributions
- Calculating Confidence Intervals
- Correlating Data
- Ranking and Percentiles
- Performing Statistical Tests
- Conducting ANOVA
- Generating Random Numbers
- Sampling Data

Time Series Analysis

- Introduction
- Plotting Time Series Data
- Adding Trendlines
- Computing Moving Averages
- Smoothing Data Using Weighted Averages
- Centering Data
- Detrending a Time Series
- Estimating Seasonal Indices
- Deseasonalization of a Time Series
- Forecasting
- Applying Discrete Fourier Transforms

Mathematical Functions

- Introduction
- Using Summation Functions
- Delving into Division
- Mastering Multiplication
- Exploring Exponential and Logarithmic Functions
- Using Trigonometry Functions
- Seeing Signs
- Getting to the Root of Things
- Rounding and Truncating Numbers
- Converting Between Number Systems
- Manipulating Matrices
- Building Support for Vectors
- Using Spreadsheet Functions in VBA Code
- Dealing with Complex Numbers

Curve Fitting and Regression

- Introduction
- Performing Linear Curve Fitting Using Excel Charts
- Constructing Your Own Linear Fit Using Spreadsheet Functions
- Using a Single Spreadsheet Function for Linear Curve Fitting
- Performing Multiple Linear Regression
- Generating Nonlinear Curve Fits Using Excel Charts
- Fitting Nonlinear Curves Using Solver
- Assessing Goodness of Fit
- Computing Confidence Intervals

Solving Equations

- Introduction
- Finding Roots Graphically
- Solving Nonlinear Equations Iteratively
- Automating Tedious Problems with VBA
- Solving Linear Systems
- Tackling Nonlinear Systems of Equations
- Using Classical Methods for Solving Equations

Numerical Integration and Differentiation

- Introduction
- Integrating a Definite Integral
- Implementing the Trapezoidal Rule in VBA
- Computing the Center of an Area Using Numerical Integration
- Calculating the Second Moment of an Area
- Dealing with Double Integrals
- Numerical Differentiation

Solving Ordinary Differential Equations

- Introduction
- Solving First-Order Initial Value Problems
- Applying the Runge-Kutta Method to Second-Order Initial Value Problems
- Tackling Coupled Equations
- Shooting Boundary Value Problems

Solving Partial Differential Equations

- Introduction
- Leveraging Excel to Directly Solve Finite Difference Equations
- Recruiting Solver to Iteratively Solve Finite Difference Equations
- Solving Initial Value Problems
- Using Excel to Help Solve Problems Formulated Using the Finite Element Method

Performing Optimization Analyses in Excel

- Introduction
- Using Excel for Traditional Linear Programming
- Exploring Resource Allocation Optimization Problems
- Getting More Realistic Results with Integer Constraints
- Tackling Troublesome Problems
- Optimizing Engineering Design Problems
- Understanding Solver Reports
- Programming a Genetic Algorithm for Optimization

Introduction to Financial Calculations

- Introduction
- Computing Present Value
- Calculating Future Value
- Figuring Out Required Rate of Return
- Doubling Your Money
- Determining Monthly Payments
- Considering Cash Flow Alternatives
- Achieving a Certain Future Value
- Assessing Net Present Worth
- Estimating Rate of Return
- Solving Inverse Problems
- Figuring a Break-Even Point

Index

Excel Scientific and Engineering Cookbook (Cookbooks (OReilly))

ISBN: 0596008791

EAN: 2147483647

EAN: 2147483647

Year: N/A

Pages: 206

Pages: 206

Authors: David M Bourg

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