Scientific Computing

This module estimates the area of the unit disk using the Monte Carlo method to approximate the integral of a function of two variables. The area of the unit disk is equal to the volume of a cylinder of height 1 having the unit disk as its base. That volume can be calculated as the integral of a function of two variables that takes the value 1 for points inside the unit disk and the value 0 for points of the circumscribing square outside the unit disk. This module applies the Monte Carlo method to approximate the integral of such a function.

The user first selects the number of sample points to be taken at a
time (1 is the default). The user then clicks on *Sample*
repeatedly to sample the circumscribing square at an additional set of
randomly chosen points. The location of each sample point is shown in
the graph by a small plus symbol. The symbol is red if the function
value is 1 (the sample point is inside the disk) and blue if the value
is 0 (the sample point is outside the disk). New samples are
distinguished from previous samples by a darker shade. Numerical
values are printed below for the correct integral *I*, the
cumulative number of samples *N*, and the current estimate of the
integral *Q* along with its error. The integral can be seen to
converge very slowly to *π* (the area of the unit disk) as the
number of samples increases.

**Reference:** Michael T. Heath, *Scientific Computing,
An Introductory Survey*, 2nd edition, McGraw-Hill, New York,
2002. See Section 8.4.4 and Chapter 13.

**Developers:** Evan VanderZee and Michael Heath