# What is a double integral?

1. Oct 24, 2012

### James2

How does this work? Like, is it integrating the integral of f(x)? Kind of like... a higher order integral? I've seen these problems before, kind of confusing; Lol random thought: InteCeption.(Also, how do I add upper and lower limits to integrals with your forum math code thing?)

$$\int \int f(x) dx dy$$

Last edited by a moderator: Oct 25, 2012
2. Oct 24, 2012

### FOIWATER

The way to carry out a double integration is to integrate f(x) first with respect to x. Then, the inside integral sign will have limits of integration for variable x. They go in for x.

Then take this function (once evaluated at the limits of integration for x) and integrate the expression with respect to variable y, and lastly evaluate it at it's limits.

Just as in the single variable case the double definite integral give you a number, and a good check is if you get a function (containing variables) you probably messed up the order of integration. Remember however that the variables can be switched to suit the problem if it is difficult to start with a certain integration.

So the basic concept is to remember to do the inside integral first, evaluate it, then do the outside. They can be switched to suit the difficulty of the problem.

For a more geometric meaning of the double integral I won't type it some one might, but this is basically how you evaluate them.

3. Oct 25, 2012

### chiro

Hey James2 and welcome to the forums.

The idea for higher integrals is that instead of summing up rectangles like you do with the Riemann integral, you are adding up rectangular solids (3D rectangular prisms) for a 2D integral and you generalize this behaviour with each new integral term.

Take a look at the link and look at the graphic:

http://en.wikipedia.org/wiki/Riemann_integral

The only difference is that you aren't looking at rectangles in two dimensions, but rectangles in higher dimensions.

Also you should be aware that the integration region isn't just a simple interval: you can integrate over really complex regions like say a circle in R^2 but the idea of adding up all these little rectangles in N dimensions is still the same.

4. Oct 25, 2012

### mathwonk

the definition is in terms of summing and taking limits of rectangular solids, but the basic theorem (fubini) says you can do it inductively by iterated integrals, i.e. repeated sums of plane rectangles.

5. Oct 25, 2012

### RedSonja

Write: \int_{lower}^{upper}f(x)dx between sets of square brackets with itex and /itex respectively or between <backslash>begin{equation} and <backslash>end{equation}. The former gives

\int_{lower}^{upper}f(x)dx

in-line, the latter creates a formula on a new line.

Last edited: Oct 25, 2012