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
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
Use the change of coordinates
$\displaystyle u=yz \quad v=xz \quad w =xy$
Solving for x, y and z gives
$\displaystyle x=\sqrt{\frac{vw}{u}} \quad y=\sqrt{\frac{uw}{v}} \quad z=\sqrt{\frac{uv}{w}} $
This has Jacobian
$\displaystyle J=\begin{vmatrix}0 & z & y \\ z & 0 & x \\ y & x & 0 \end{vmatrix}=2xyz$
So this gives that
$\displaystyle dudvdw=2xyzdxdydz \iff dxdydz=\frac{dudvdw}{2\sqrt{uvw}}$
Now you have the integral
$\displaystyle \iiint 1 dxdydz= \int_{a^2}^{b^2} \int_{a^2}^{b^2} \int_{a^2}^{b^2} \frac{dudvdw}{2\sqrt{uvw}}$