Given the minor and major axes of an ellipse, it is easy to find the formula that describes its area (the perimeter is more difficult). See this link. I would like to know if we can use calculus to find the area of an ellipse when we know the Cartesian coordinates of the foci, call them $(a_1,a_2)$ and $(b_1,b_2)$, and the value $c$ so that
$\sqrt{(x-a_1)^2+(y-a_2)^2} + \sqrt{(x-b_1)^2+(y-b_2)^2} = c$ (the distance function)
As shown in the second answer to this question, we can determine the length of the minor and major axis of the ellipse with the information that we have, and hence use this to find the area of the ellipse. What I'm looking for is an alternate method that does not consider the major and minor axis of the ellipse (perhaps by taking a double integral?) and gives the area of the ellipse with respect to the foci of the ellipse. I would be interesting in seeing such a method and I would like to know: when we are given an implicit function such as the one above that describes a simple closed curve, can we in general always find the area of its interior region by using calculus?
Edit: (removed earlier edit) Note that, if I'm not mistaken, we can rewrite the above equation as
$c^4 – 2c^2((x-a_1)^2+(y-a_2)^2+(x-b_1)^2+(y-b_2)^2) + ((x-a_1)^2+(y-a_2)^2-(x-b_1)^2-(y-b_2)^2)^2 = 0$
It is possible to solve the above equation for $y$, however the result takes for too much space to copy here (at least the result that I got on WolframAlpha).
Best Answer
This is easily solvable with resultants. Just set up the parameters for the coordinates of the foci and c, then write down five pretty obvious polynomial equations, then use the Dixon resultant. Here's the answer (assuming I didn't make a dumb mistake):
$A = \pi*c*s/4$
where
$s = \sqrt{ c^2 - (f_2y-f_1y)^2 - (f_2x-f_1x)^2 }$
The foci are $(f_1x, f_1y)$ and $(f_2x, f_2y)$.
Note that this reduces to the right thing if the foci are equal, or if c = the distance between the foci.