Question #40513

The Argand plane $\mathbb{C}$ is embedded in $\mathbb{R}^3$ as the plane $\Pi: z = \sqrt{3}x$. The origin of $\mathbb{C}$ maps to $(0,0,0)$. The real axis of $\mathbb{C}$ coincides with the line of intersection of $\Pi$ and the $xz$-plane, with the positive real direction having positive $x$. The imaginary axis of $\mathbb{C}$ coincides with the $y$-axis. Let $R$ be the region in $\mathbb{C}$ defined by the inequalities $|w| \le 2$ and $\log_{|w|}(|w - 2|) \le 1$. A solid $S$ in $\mathbb{R}^3$ is bounded below by the $xy$-plane ($z=0$), above by the plane $z = 2\sqrt{3}x + 5$, and laterally by the vertical cylinder passing through the boundary of the orthogonal projection of $R$ onto the $xy$-plane. Find the volume of the solid $S$.