Testwiki:Reference desk/Archives/Mathematics/2012 July 12

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July 12

Area of part of an oval

File:For RD question relating to area of part of an oval.gif

What are the areas of A_L and A_S?

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)

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 $\underset{-r}{\overset{r}{\int }}|y-s|dx$ 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)

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

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)

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)

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)

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)

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