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I was trying to calculate and find out the area of a sector of an ellipse when the angle of the sector is drawn from one of the focii and one arm is taken as the x-axis.

I found the following question asked a decade ago: How to calculate ellipse sector area from a focus

Top answer got 9 upvotes which uses the values of some of the components of ellipse such as the semi-minor axis, semi-major axis, eccentricity, eccentric anomaly, mean anomaly to derive the formula $$\tfrac12ab\left(E-e\sin E\right)$$ i.e. $$\tfrac12abM$$

But I am more interested in the next answer which got 7 upvotes and somehow derived the formula: $$\int_{\theta_1}^{\theta_2}\frac{1}{2}r^2d\theta,$$ where $r=r(\theta)$ is the equation of the ellipse, with polar origin at the focus.

Can somebody please help me and explain how the formula was derived along with suitable example?

Because I am unable to figure out the derivation myself and when I tried to use both the formula to find sector area in same ellipse, I got some discrepancy like very different values. So, I am confused but I am assuming that both formulas are correct as they got almost equal upvotes.

Thanks in advance and edits are always welcome!

See the answer here: https://math.stackexchange.com/a/388155/1379223

Math_Maven
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    That's the standard formula for the area in polar coordinates. – Intelligenti pauca Oct 07 '24 at 07:57
  • I see, any idea then why I might be getting different values even when used for the same sector of same ellipse? – Math_Maven Oct 07 '24 at 17:34
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    @Intelligentipauca Thanks a lot! Searching "standard formula for the sector area of ellipse in polar coordinates" on the web landed me to "https://math.libretexts.org/Bookshelves/Calculus/Map%3A_Calculus__Early_Transcendentals_(Stewart)/10%3A_Parametric_Equations_And_Polar_Coordinates/" which explains a lot about "Areas and Lengths in Polar Coordinates" in detail – Math_Maven Oct 07 '24 at 17:40

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