Well, that was cool if not tedious but I understand the Jacobian and its application to changing coordinate systems. $${J_{POLAR}= \rho}$$ $$ {J_{cyl}= \rho}$$ and $${J_{sphere}=\rho^2\sin\phi}$$ Let me see if I have this right any time I find the area of a polar conic I take evaluate the integral like so $${\int_0^R\int_c^d \rho d\rho d\theta}$$ and similarly for volume in cylindrical and spherical respectively $${\int_0^R\int_o^\theta\int_0^z \rho \theta d\rho d\theta dz}$$ $${\int_0^R\int_0^\theta\int_0^\phi \rho^2\sin\phi d\rho d\theta d\phi}$$
Say for a conic in polar $$ {\rho = (ed/(1-e\sin\theta)}$$
Ok I just hit my next road block but let me take a shot in the dark the area under the conic would be??
I am lost here, but I think I get $${\int_0^R\int_0^\theta \rho[ (ed)/(1-e\sin\theta)]d\rho d\theta}$$
Uh.. that just does not look right..what am I missing here?