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Let $\Omega:=\{(x,y,z)|x^2+y^2=1, 0\leq z \leq 2\}$, fix an $\alpha \in (-\frac{\pi}{2},\frac{\pi}{2})$ and given the transoformation $T(x,y,z):=(x,y+z\tan \alpha,z)$, find the volume of $T(\Omega)$.

Remark: By the coordinate transformation formula for multiple integrals one can compute the corresponding Jacobian to be $1$, thus the volume of $T(\Omega)$ is the same as the volume of $\Omega$, am I right?

Also, geometrically, the transformation only shifts the $xy$-cross section (at each $z$ level) to an extent (along the $y$ axis), thus the area of each cross section of the cylinder $\Omega$ remains unchanged after tranformation, so the volume of $T(\Omega)$ is equal to the one of $\Omega$.

Is my explaination correct?

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up vote 1 down vote accepted

Yes, the determinant of the (Jacobian) derivative tells the extent how the volume changes, and it is indeed $1$ in all points at this case ('tilting', or 'shearing'), so the volume stays the same.

You can also convince yourself geometrically: by approximating the volume of the straight cylinder by small cubes (or rectangular solids), and then apply $T$, and a simple draw shows clearly that the tilted cube has the same volume as the orginal cube.

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