The double integral (or area integral) is calculated for simply connected areas in the xy plane. The integrand is a function f(x,y).
Definition of the integration area
If the integration area is bordered by a polyline, then
this area can be defined by a list of points P(x_{i}/y_{i}).
The boundary points P(x_{i}/y_{i}) can be edited as a table or moved with the mouse.
The third column of the table is not necessary for these points.
Curvilinear boundaries can be included approximately by providing boundary curves in parameter form.
The curve parameter is always t. For example, if a limit curve is given as y=f(x), we can assign x(t)=t and y(t)=f(t).
For the curve parameter t, the beginning and end of the respective edge curve segment are specified in the third column with two entries.
When choosing the order of the two t values, it is important to ensure that each curve segment for t_{1}->t_{2} is run through in the same direction as all the others.
This means that all curves must encircle the integration area clockwise or counterclockwise.
If there is a gap between 2 consecutive curve definitions, a straight connection is automatically assumed.
Numerical methods
There are 2 numerical methods to choose from: Monte Carlo Integration and Gaussian Integration.
For Gaussian Integration, the area is divided into small triangular areas and f(x,y) is evaluated at 3 points each.
"Stratified sampling" is used in Monte Carlo Integration, that is, the area is divided into small rectangular areas of equal size, in which f(x,y) is evaluated at a chosen point each one at random.
Application
The most obvious application of the double integral is the calculation of a volume.
If f(x,y)≥0 for the integrand f(x,y) of the double integral in the entire region of integration,
then the double integral gives the volume of the columnar body bordered by the xy plane at the bottom and
by the surface z=f(x,y) at the top. The lateral limit is given by the limit curve of the area.
Thus, a double integral with integrand 1 gives the volume of a disk of thickness 1 bounded by the edge curve or the area of the region of integration.
If you compute a double integral with integrand x and divide it by the area of the region of integration, you get the x coordinate of the centroid of the region of integration.