Volume after transformation.

Question

Consider $$ \Omega:\{ (x, y, z): x^2+y^2\leq 1, 0\leq z\leq 2 \} $$ and the transform $$ T:(x, y, z)\to(x, y+\tan(\alpha z), z) ,$$ where $ \alpha\in (0, \pi) $. What is the volume of $ T(\Omega) $?

My attempt: I am trying to convert to the cylindrical coordinate system, but I can't find the region where I should integrate since it is transformed.

Answer 1

Fix $z_0 \in [0,2]$, and look at the section $\Omega_{z_0} = \Omega \cap \{z = z_0\}$. This section is mapped to $$ T(\Omega_{z_0}) = \{ (x,y,z_0) : x^2 + (y - \tan(\alpha z_0))^2 \leq 1\} $$ which is a unit circle in the plane $\{z = z_0\}$, centered at $(0, \tan (\alpha z_0))$. Hence, by the Cavalieri Principle, $T(\Omega)$ has the same volume as $\Omega$.

Answer 2

Since the Jacobian for the transformation is

$$|J|=\begin{vmatrix}1&0&0\\0&1&0\\0&\frac{\alpha}{\cos^2{\alpha z}}&1\end{vmatrix}=1$$

for $\alpha\neq \frac{\pi}2$ we have have that

$$V_{T(\Omega)}=V_{\Omega}=2\pi$$