Change of Variables in a 3 dimensional integral

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.

-
Hint: try calculating the volume of a sphere in Cartesian and in Spherical coordinates. – Jerry Schirmer Mar 1 at 20:52