Page 377 - Excel for Scientists and Engineers: Numerical Methods

354 EXCEL: NUMERICAL METHODS

Monte Carlo Integration
The Monte Carlo method can be used to integrate a function that is difficult
or impossible to evaluate by direct methods. Often the process of "integration" is
the determination of the area of a figure. We'll illustrate the technique by
determining the area of two figures: first, the area of a circle (from which we can
evaluate n), and second, the area of an irregular figure.
The evaluation of x is a classic illustration of the determination of an area by
the Monte Carlo method. Two random numbers in the range -1 to +1 are used to
determine the coordinates of a point in the x, y plane. The number of points
inside the circle, defined by the equation x2 + y2 = 1 , divided by the total number
of points, gives the ratio of the circle to the circumscribing square. Figure 15-14
illustrates such a calculation, using 4000 points.

Figure 15-14. Estimation of 7c by using RAND.

This particular calculation gave 3.129 as the value of x.

The Area of an Irregular Polygon
When the preceding method is used to estimate the area of an irregular
figure, we need a general method to determine whether a given point is inside or
outside the figure. In the following, the figure must be a polygon, that is, a figure
that can be described by a series of coordinates connected by straight lines.
Since in an Excel chart, a curve can be approximated by a number of straight line
segments, in theory a figure of any shape can be handled.