TITLE: Deriving the Area of a Sector of an Ellipse
QUESTION [12 upvotes]: A sector $P_1OP_2$ of an ellipse is given by angles $\theta_1$ and $\theta_2$. 

Could you please explain me how to find the area of a sector of an ellipse?

REPLY [2 votes]: As I show in Evaluating $\int_a^b \frac12 r^2\ \mathrm d\theta$ to find the area of an ellipse the area 
of an ellipse given its central angle is: 
$A(\theta) = \frac{1}{2} a b \tan ^{-1}\left(\frac{a \tan (\theta )}{b}\right)$
for $0\leq\theta\leq2\pi$, so the area of the sector in question is: 
$A\left(\theta _2\right) - A\left(\theta _1\right)$ 
or: 
$\frac{1}{2} a b \left(\tan ^{-1}\left(\frac{a \tan \left(\theta 
_2\right)}{b}\right)-\tan ^{-1}\left(\frac{a \tan \left(\theta 
_1\right)}{b}\right)\right)$ 
which sadly does not appear to simplify any further.