Consider $$R=\{(x,y,z):x^2+y^2\leq 1,0\leq z\leq 2\}$$
and the transform $$T:(x,y,z)\to(x,y+\tan \alpha z,z)$$
where $0<\alpha<\pi$
Then what is the volume of $T(R)$?
I tried myself but I confused because of the fourth parameter $\alpha$,I do not know how to solve this.
This is a linear transformation, so by definition, the Volume will be the your old volume scaled by the determinant of the transformation.
The starting point is clearly a cylinder of volume $4\pi$. If you look at the associated matrix, you'll find that the determinant of the matrix is 1, so the volume is unchanged at $4\pi$. You can probably try to understand what's going on better if you study the transformation in the plan $(y,z)$.