Finding Areas with the Gauss-Green Formula

                        Copyright 1994
                       H. Edward Donley
                    Mathematics Department
                   Indiana University of PA
                    Indiana, PA  15705 USA
                    hedonley@grove.iup.edu

Introduction

Parameterizing the Boundary

Plot[{Sqrt[1-x^2],-Sqrt[1-x^2]}, {x,-1,1},
  AspectRatio->Automatic];

Click on the graph. A box will appear enclosing the graph. Now hold down the Command key (the one with the picture of an apple on it. If you move the mouse cursor into the graph's bounding box, the cursor will turn into a cross hairs and the coordinates of the point will appear in the bottom left corner of the notebook window. Hold down the mouse button and click on the circle. Continue to keep the mouse button depressed and click around the circle counterclockwise until you get back to where you started. Now release the mouse button. Select Copy under the Edit menu. Click in the following input cell and then select Paste under the Edit menu. You should see a list of all of the data points that you selected. Evaluate the input cell to assign the list of data points to the variable, data.

data = ***paste data here***;

dataLoop = Append[data, First[data]];

Let's take a look at these points to verify that they go around the circle.

ListPlot[data, AspectRatio->Automatic];

Good enough? If not, you can always start over again and pick a new set of data points.

Now we separate out the x and y values.

x = Transpose[dataLoop][[1]];
y = Transpose[dataLoop][[2]];

Numerical Approximation Method

to this data list. The boundary of the region can be divided into segments whose endpoints are the data points that you picked off of the graph. We will denote the x coordinate of the ith point by x_i and the y coordinate by y_i. Segment i connects {x_i, y_i} and {x_(i+1),y_(i+1)} for i varying from 1 to the number of data points. On segment i, we will approximate x[t] with the average of the two endpoints, 0.5 (x_(i+1) + x_i). Similary we will approximate y[t] with 0.5 (y_(i+1) + y_i). We will approximate y'[t] dt with the change in y, y_(i+1) - y_i, and x'[t] dt with x_(i+1) - x_i. So we can approximate the area with

and here is the Mathematica code.

area =
   0.5 Apply[
      Plus,
      Table[
         0.5(x[[i+1]]+x[[i]]) (y[[i+1]]-y[[i]])
        -0.5(y[[i+1]]+y[[i]]) (x[[i+1]]-x[[i]]),
      {i,1,Length[data]}]
   ]

Did your answer make sense? What should the area be?

Exercises

Exercise 1: Area of Russia