Double Integration.

WORK IN PROGRESS

Let’s start the blog with the definition of a Double integration and some examples.

An Integral is defined as the sum of all the infinitely small diferentials of the surface/area... or as the area of the region in the xy-plane bounded by the function F and the x-axis. It may also be referred as the "Antiderivative" due to the fact that is the opposite of a Derivate.

In the case of the double integral, the double integral of a positive function of two variables (x and y; Ro and Theta) represents the volume of the region between the surface defined by the function (on the three dimensional Cartesian plane where z = f(x, y) and the plane which contains its domain. 

The result of a double Integral is obtained in a similar way of the Definite Integral, but following some steps. First you have to decide which of your two variables is going to be the Main Variable, and which is going to be the Auxiliary Variable. Once you have done this, you have to integrate first the Auxiliary variable, treating the other variable as a constant, and then, once you have applied the values to the integrated function, you integrate a second time with the Main variable and apply its values.

Lets see an example so we can see it more visually.

Function F(x,y)= xy^2

In a double integral, the outer limits must be constant, but the inner limits can depend on the outer variable. This means, we must put y (the Auxiliary variable) as the inner integration variable, as shown in the following image:

There are many ways to solve a Double Integral, depending on which Geometrical Figure we are dealing with and in with plane (Cartesian, Polar or Bipolar) this figure is, but we will always solve this integrals using this same method, but changing the variables for each coordinate change.

Hope you have enjoyed this post and Hope to receive some feedback on what I can improve.

Here I will leave you some links so you can learn more about Double Integration.

Paul´s Online Math Notes

Video: Calculating a Double Integral.   

(Best of all Youtube Videos even though is on moodle)

I DO HAVE TO CHANGE THIS POST IN ORDER TO MAKE ALL THE BLOG ABOUT THE SAME SUBJECT.