I am working on a paper and I wish to use this equation for calculating the thrust of an engine in a nacelle: $$ Fn= \displaystyle \oint V_x\rho\vec V\cdot\vec n \ dA + \displaystyle \oint(P-P_{\infty})\vec n_x \ dA - \displaystyle \oint \vec \tau_x \ dA$$
This equation was given for a simplified case: the common circular nacelle. However, I would like to apply it to the flattened bottom nacelle as on the 737, as shown here:
I have attempted to fit an equation to this by adjusting the parameters of a cardiod in polar coordinates. I'm having some trouble changing the above to polar integrals, however I'm not sure as to how this conversion would affect the rest of the equation. I could apply the rectangular form instead and keep the above integrals, but I don't think there is a away to express y explicitly in terms of x; it must be done implicitly and so that integral wouldn't really be possible.