How do I find the area between 3 curves? I have three equations: $y=3/x$, $y=12x$, and $y=x/12$, $x>0$. I am not sure how to go about integrating an equation once I find the intersections. Do I need multiple integrals?

Yes. After drawing a diagram, you should be able to see that the desired area is:
$$
\int_0^{1/2} (12x - \tfrac{x}{12}) \, dx + \int_{1/2}^6 (\tfrac{3}{x} - \tfrac{x}{12}) \, dx
$$
