# Double integral with change of variables.

Find the integral of f(x,y)=x^2 over the domain D which is bounded by y=3x, x=3y and x+y=4

Hint: use the transformation x=3u+v and y=u+3v

https://brainmass.com/math/integrals/double-integral-change-variables-26210

#### Solution Preview

Hi there

Here are the solutions this problem.

I solved it both using the change in variables and by brute force.

I was shocked - shocked I tell you - to see that I got the same results in both methods.

To further my amazement, even Maple agreed with me :)

to tell you the truth, the change of variables here is a pain in the neck. The Brute force method was much more elegant in this case.

Ok, if they give us a hint, then we better use it.

We want to convert this integral to:

There are three steps to solve this ...

#### Solution Summary

The solution shows how to change the integration variables and limits to simplify the integration. In addition the solution shows how to solve the integral directly and get to the same result.

The solution is 7 pages long including diagrams and full derivations.