Units of FFT frequency
The formula you made will work if what you are after is simply "the area under the the magnitude plot from $\omega_i$ to $\omega_f$."

In many cases you can think of Hz as meaning "rotations per second."  In that case, it becomes more obvious that 1 rotation = 2$\pi$ radians.  Since $f$ has units of Hz and $\omega$ has units of radians, the conversion is simply:
$$
1 \, \text{Hz} = 2 \pi \, \frac{\text{rad}}{\text{s}}
$$
So evaluate your integration from $\frac{1}{2\pi}$ to $\frac{10}{2\pi}$ and you should have the result you are after.