I'm working through a geometrical problem where I need to find the area on the 2D plane between different values of $L$, where:
$$L = \sqrt{(A^2 + x^2 + y^2)} + By\tag1$$
I'm struggling to find an analytic solution to the area of these weird ellipses, here's an example of what they look like. The regular ellipse equation is:
$$1 = \frac{x^2}{a} + \frac{y^2}{b}$$ Or, by setting $L = ab$, one could write:
$$L = bx^2 + ay^2\tag2$$ $a$ and $b$ are the semi-major and semi-minor axis and the area of this ellipse is: $$\text{Area} = \pi ab $$ But I cannot find a way to put $(1)$ into the form of $(2)$ nor can I find a semi-major or semi-minor axis for this shape so I'm a bit stuck on finding the area. I also tried finding the area using integration over some infinitesimal $dL$ and $d\theta$ angle but couldn't work out how to integrate them.