How to calculate ellipse sector area *from a focus* How do you calculate the area of a sector of an ellipse when the angle of the sector is drawn from one of the focii?  In other words, how to find the area swept out by the true anomaly?
There are some answers on the internet for when the sector is drawn from the center of the ellipse, but not from the focii.

The area will be 
$$\int_{\theta_1}^{\theta_2}\frac{1}{2}r^2d\theta,$$
where $r=r(\theta)$ is the equation of the ellipse, with polar origin at the focus.
Imagine an ellipse with semi-major axis $a$ and eccentricity $e$, and with one of the foci at the origin, and the other focus on the half-line $\theta=0$ (so to the "right" of the origin). Then the ellipse has polar equation
$$r=\frac{a(1-e^2)}{1-e\cos\theta}.$$
The rest is unpleasant. There is a closed form for the antiderivative. One might ask Wolfram Alpha. If you want to do it yourself, you can use the Weierstrass substitution. But there is undoubtedly a neater way. 
