Complex City getting bigger

The Cauchy Integral takes on the form $\frac{1}{2\pi i} \cdot \oint_C \frac{f(z)}{z - a} dz$ , where $C$ is the unit circle centered at the origin of the complex plane. The above complex integrand can be rewritten as:

$\frac{z^3 - 6}{3z - i} = \frac{z^3 - 6}{3(z - i/3)} = \frac{z^{3}/3 - 2}{z - i/3}$

The pole $z = \frac{i}{3}$ is contained within $C$ and thus the Cauchy Integral computes to $2i\pi f(a) = 2i\pi f(i/3) = 2i\pi \cdot (\frac{1}{3} \cdot (\frac{i}{3})^3 - 2) = \boxed{\frac{2\pi}{81} - 4i\pi}.$