Homework Help: Order of integration.

1. Dec 18, 2011

Kuma

1. The problem statement, all variables and given/known data

I have this question given:

2. Relevant equations

3. The attempt at a solution

So I used change of variables, fairly straightforward, I set

a = x+y+z
b = x+2y
c = y - 3z

computed the jacobian, and got the new ranges.

Anyway, so the solution has the order of integration as
int int int (sqrt a dc db da)

why is it from dc db da? I was wrong at that part because i used da db dc, how do you find out what the order of integration should be?

2. Dec 18, 2011

Markus Hanke

What you can do is to regard the boundary conditions as a system of linear equations. This will allow you to calculate values for x,y and z ( this is where the parallelpipes intersect ). Now stay in the same coordinate system and simply integrate over dxdydz using the values obtained earlier - make sure they are oriented correctly, i.e. the signs are correct. The actual integration should then be straightforward.

3. Dec 18, 2011

Kuma

I get what you are saying, but its volume, so why does it matter if you integrate with respect to z, y and x rather than in the other order? (assuming the restrictions are correct for each order).

Change of variable has to be used for this question because it makes it a lot easier. What I want to know though is the order of integration when you perform the change of variable.

4. Dec 18, 2011

Markus Hanke

Well, the end result, if done correctly, will be the same for each method.
I personally think that this particular integral is actually much easier to calculate if you stay in the [x,y,z] space; the order of variables doesn't need to change either, in fact the order doesn't even matter so long as you have the integration limits right.