Use a change of variables to transform the above integral to an integral over the cube $ \ \{0<u<1, \ 0<v<1, \ 0<w<1 \} \ $

Question

Consider the tripple integral $ \ \large \int_{2}^{3} \int_{4}^{8} \int_0^{10} \sin (xy-z) dxdydz \ $

Use a change of variables to transform the above integral to an integral over the cube $ \ \{0<u<1, \ 0<v<1, \ 0<w<1 \} \ $

Answer:

I am having the problem to find the required transformation from the given domain to the unit cube.

The given domain is

$ \ 0 \leq x \leq 10, \\ 4 \leq y \leq 8 , \\ 2 \leq z \leq 3 \ $

We have to find a transformation $ \ u=\phi(x,y,z) , \ v=\psi(x,y,z), \ w=\eta(x,y,z) \ $ that maps the given domain into the unit cube $ \ \{0<u<1, 0<v<1 , 0<w<1 \} \ $

Let us consider the transformation $ u=\frac{1}{10}x(x-9) , \ v=\frac{1}{4} y-1 , \ w=z-2 \ $

Will this transformation work ?

But I think this will make complicated when subsituted in the integral.

Any other easy transformation keeping in mind the integrand $ \ \sin (xy-z) \ $ will be helpful

Any help is appreciated

Answer 1

Hint: Consider the transformation $$ \begin{pmatrix}x\\y\\z\end{pmatrix}\mapsto \begin{pmatrix}\frac{x}{10}\\\frac{y-4}{4}\\z-2\end{pmatrix} $$