Multivariable Calculus: Change in Variables and Volume

The question:

"Let A be the region defined by the inequalities:

x > 0, y > 0, z > 0,

1 ≤ xyz ≤ 8,

x ≤ y ≤ 2x,

x ≤ z.

(a) Find the volume of A.

(b) Find the integral of e^(xyz) over A."

____________

What I've tried:

I'm not even sure what the right substitution to use here is. I was thinking u = xyz, v = y/x, w = z/x which would give us

1 ≤ u ≤ 8,

1 ≤ v ≤ 2

1 ≤ w

and I was thinking that we could compute the Jacobian from this (though it would be quite messy...) and then use these intervals as the bounds for u,v,w when we integrate in the order du dv dw.

Re: Multivariable Calculus: Change in Variables and Volume

Hey twilightmage13.

I would try that approach and see how you go.

Since you are dealing with mostly functions that are easy to differentiate partially you should get a Jacobian a lot easier than you think.

For these kinds of problems (triple or higher dimensional integrals) you may have to play around with things to get enough insight to know what is going on.

Re: Multivariable Calculus: Change in Variables and Volume

Thank you - but my biggest problem now is how do we determine the bounds of the triple integrals?