Change of Variables in a 3 dimensional integral

Question

Let $\int_0^{\infty}\int_0^{\infty}\int_{-\infty}^{\infty}f(x_1,x_2,x_3)dx_1dx_2dx_3$ be a 3 dimensional variable ( i.e. $0\leq x_1,x_2\leq \infty,-\infty\leq x_3\leq \infty.)$ I am defining the following change of variables : $$x_1'=x_1,x_2'=x_2, x_3'=c_1x_1+c_2x_2+c_3x_3$$

Question : What will be my new domain expressed in $x_1',x_2',x_3'.$ I understand the Jacobian concept but it is not clear how to define my new boundaries.

Answer 1

The boundaries transform the same as the variables.

Answer 2

New limits coincide with the old ones.