jmc

Let $z$ be a member of the set $T$. Then $z=w-\frac{1}{w}$ for some complex number $w$ with absolute value $3$. We can rewrite $z$ as $$z=w-\frac{1}{w}=w-\frac{\overline{w}}{|w{|}^2}=w-\frac{\overline{w}}{9}.$$Let $w=x+iy$ where $x$ and $y$ are real numbers. Then we have $$z=x+iy-\frac{x-iy}{9}=\frac{8x+10iy}{9}.$$This tells us that to go from $w$ to $z$ we need to stretch the real part by a factor of $\frac{8}{9}$ and the imaginary part by a factor of $\frac{10}{9}$.

$T$ includes all complex numbers formed by stretching a complex number of absolute value $3$ in this way. Since all complex numbers of absolute value $3$ form a circle of radius $3$, $T$ is an ellipse formed by stretching a circle of radius $3$ by a factor of $\frac{8}{9}$ in the $x$ direction and by a factor of $\frac{10}{9}$ in the $y$ direction. Therefore, the area inside $T$ is $$\frac{8}{9}\cdot \frac{10}{9}\cdot 9\pi =\boxed{\frac{80}{9}\pi }.$$