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July 12

Area of part of an oval

If I take an oval and cut it along a line that is parallel to its minor axis but not on its minor axis, how can I work out the area of the two pieces? Handschuh-^{talk to me} 01:33, 12 July 2012 (UTC)[reply]

The ellipse has the equation

{\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1

You can solve this for y, at least for the smaller of the two pieces. Then you can calculate the area as the integral

\int _{-r}^{r}|y-s|dx

for s corresponding to where you place your cut line and r chosen such that y(r)=s. The resulting integral is be possible to calculate by elementary techniques (unlike the arc length). —Kusma (t·c) 07:42, 12 July 2012 (UTC)[reply]

See List of integrals of irrational functions. You should end up with an arcsin function. — Quondum ^☏ 08:13, 12 July 2012 (UTC)[reply]

An ellipse is a circle with a scale factor greater than unity in the direction of the major axis. Such a transformation doesn't change relative areas, so the ratio of your two areas will be the same as for a circle (diameter equal to the ellipse's minor axis) with a cut of the same length.→86.139.64.77 (talk) 07:49, 12 July 2012 (UTC)[reply]

That should also work. See circular segment for the formulas valid for the circle (note that the scaling will change the angles). —Kusma (t·c) 08:44, 12 July 2012 (UTC)[reply]

So if I take my scale factor as b/a, and work out the area of the segment for a circle of diameter a, then that area times the scale factor will be the area A_S? Handschuh-^{talk to me} 08:48, 12 July 2012 (UTC)[reply]

Yes. Equivalently, ignore the scale factor and use the ratio of the areas, as that's the same in both the ellipse and the circle as stated by 86.139.64.77. — Quondum ^☏ 12:47, 12 July 2012 (UTC)[reply]

Thanks for your help! Handschuh-^{talk to me} 03:39, 13 July 2012 (UTC)[reply]

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