Volume integral, how do I find the limits for my integral?

**LHS** · May 17th 2011, 02:11 PM

Hi, does anyone know how to find the integral that needs to be evaluated here? I can't understand how to find it from the region

thanks in advance!

from here, not take home test, sheet 3
http://www.maths.ox.ac.uk/courses/course/12489/material

**TheEmptySet** · May 17th 2011, 04:37 PM

Originally Posted by LHS

Hi, does anyone know how to find the integral that needs to be evaluated here? I can't understand how to find it from the region

thanks in advance!

from here, not take home test, sheet 3
Course Material for Calculus in Three Dimensions and Applications | Mathematical Institute - University of Oxford

Use the change of coordinates

$u=yz \quad v=xz \quad w =xy$

Solving for x, y and z gives

$x=\sqrt{\frac{vw}{u}} \quad y=\sqrt{\frac{uw}{v}} \quad z=\sqrt{\frac{uv}{w}}$

This has Jacobian

$J=\begin{vmatrix}0 & z & y \\ z & 0 & x \\ y & x & 0 \end{vmatrix}=2xyz$

So this gives that

$dudvdw=2xyzdxdydz \iff dxdydz=\frac{dudvdw}{2\sqrt{uvw}}$

Now you have the integral

$\iiint 1 dxdydz= \int_{a^2}^{b^2} \int_{a^2}^{b^2} \int_{a^2}^{b^2} \frac{dudvdw}{2\sqrt{uvw}}$

**LHS** · May 18th 2011, 01:12 AM

Awesome! But I am confused how you obtained that jacobian, dx/du isn't equal to zero

Edit: Ah.. I see what you're doing, thanks, that's all sorted now

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