What is the correct order of integration when performing a change of variables?

[Kuma]

Homework Statement

I have this question given:

Homework Equations

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?

 

[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.

 

[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.

 

[Markus Hanke]

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.

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.