Double Integrals: An Introduction

A general definition of a double integral: The volume between the surface, determined by a function of a function in R^3 and the plane that is determined by the domain (ie: what are to be the upper and lower limits of the integral).

In essence, we are trying to find the volume under (or above a surface) as determined by the function over a set interval, given by the upper and lower limits of an integral.

On the photos, you will see various examples of how double integration is performed.  Then with the associated number, I will provide any extra comments.  

Example 1 is just a motivating example just to understand how integrals work. Now, here, we assumed that the limits or the domain of our integration is a rectangle (as all values of the upper and lower limits were constants). Let's tweak the first example and give limits that might not have a rectangular domain over which we would integrate.

Please note that the last integration should NOT ever contain any variables as that would not give a definite value (you would most likely end up with a integration involving variables, not numerical values

There are also some times when you might have to change the order of integration to either make life a little easier or to actually make it possible to integrate. I will show a few examples of such later. Many of these types of examples deal with integration that might either require some form of integration by parts, trig substitution and u-substitution.

As stated from the diagram:

The region in orange is the region over which we want to integrate.  Note that for the limits of y (0,x^2), there are no y values, and for integration with respect to x, there are no variables. 

The last integration when using multiple integration should never contain any variables, and any other integration should not limits that contain any part of the variable to which you are integrating.

Now, we must introduce integration by changing the order of variables.  It isn't as direct as one would assume.  Before we do so, we must introduce Fubini's Theorem, which allows us to change the order of integration under certain conditions.

When we want to switch the order of integration we must acknowledge Fubini's Theorem. In summary it states that if there are no discontinuities that are greater than area 0 (ie. points, lines) and if there are no discontinuities such that each line parallel to the x or y axis will AT MOST touch the discontinuity finitely many times over the domain x⊆[a,b] y⊆[c,d], then it is possible to change the order of integration and get the same result. Note: the limits will change for each x and y (ie: if you started with integrating with respect to (w.r.t.) x, then you will now have y with functions).

We can use Example 2 to better understand this theorem.

Other places where you might be able to find some more details about math:

Khan Academy - This site has great interactive videos, and there have been great reviews about the videos.

http://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx - This site is great for an overall view of Calculus III, with a lot more details and examples.

www.omegaprep.com - Our affordable online tutoring services, with an easy to use platform.