Doppler effect on phase

I am struggling to find or derive the relationship between Doppler frequency and the associated phase shift. Let's say I have a signal transmitted at a given frequency $f_c$. I then down convert from $f_c$ to baseband at the receiver, but due to speed between the transmitter and receiver, I receive a frequency shifted by a Doppler frequency $\Delta f$.

If I wish to generate a matching signal to that being received, I can use

$$\cos(2 \pi \Delta f t + \phi) + i \sin(2 \pi \Delta f t + \phi)$$

The part I'm struggling to get to grips with is how does $\phi$ relate to $\Delta f$? Is it simply the integral of $\Delta f$ with respect to time?

$$\phi = \int_0^t \Delta f dt$$

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You are correct that the total accumulated phase offset due to Doppler could be calculated by integrating the Doppler shift $\Delta f(t)$ over time. In addition to Doppler shift, however, you would also need to take into account any frequency difference between the transmitter and receiver's reference oscillators.