Editing Ellipse (section)

=== Area ===
The [[area]] <math>A_\text{ellipse}</math> enclosed by an ellipse is:
{{NumBlk2||<math display="block">A_\text{ellipse} = \pi ab</math>|2}}

where <math>a</math> and <math>b</math> are the lengths of the semi-major and semi-minor axes, respectively.  The area formula <math>\pi a b</math> is intuitive: start with a circle of radius <math>b</math> (so its area is <math>\pi b^2</math>) and stretch it by a factor <math>a/b</math> to make an ellipse. This scales the area by the same factor: <math>\pi b^2(a/b) = \pi a b.</math><ref>{{Cite book|last=Archimedes.|url=https://www.worldcat.org/oclc/48876646|title=The works of Archimedes|date=1897|publisher=Dover Publications|others=Heath, Thomas Little, Sir, 1861-1940. | isbn=0-486-42084-1 | location=Mineola, N.Y.|pages=115|oclc=48876646}}</ref> However, using the same approach for the circumference would be fallacious – compare the [[integral]]s <math display="inline">\int f(x)\, dx</math> and <math display="inline"> \int \sqrt{1+f'^2(x)}\, dx</math>. It is also easy to rigorously prove the area formula using integration as follows.  Equation ({{EquationNote|1}}) can be rewritten as <math display="inline">y(x) = b \sqrt{1 - x^2 / a^2}.</math>  For <math>x\in[-a,a],</math> this curve is the top half of the ellipse.  So twice the integral of <math>y(x)</math> over the interval <math>[-a,a]</math> will be the area of the ellipse:
<math display="block">\begin{align}
  A_\text{ellipse} &= \int_{-a}^a 2b\sqrt{1 - \frac{x^2}{a^2}}\,dx\\
                   &= \frac ba \int_{-a}^a 2\sqrt{a^2 - x^2}\,dx.
\end{align}</math>

The second integral is the area of a circle of radius <math>a,</math> that is, <math>\pi a^2.</math> So
<math display="block">A_\text{ellipse} = \frac{b}{a}\pi a^2 = \pi ab.</math>

An ellipse defined implicitly by <math>Ax^2+ Bxy + Cy^2 = 1 </math> has area <math>2\pi / \sqrt{4AC - B^2}.</math>

The area can also be expressed in terms of eccentricity and the length of the semi-major axis as <math>a^2\pi\sqrt{1-e^2}</math> (obtained by solving for [[flattening]], then computing the semi-minor axis).

[[Image:tiltedEllipse2.jpg|thumb|The area enclosed by a tilted ellipse is <math>\pi\; y_\text{int}\, x_\text{max}</math>.]]

So far we have dealt with ''erect'' ellipses, whose major and minor axes are parallel to the <math>x</math> and <math>y</math> axes. However, some applications require ''tilted'' ellipses. In charged-particle beam optics, for instance, the enclosed area of an erect or tilted ellipse is an important property of the beam, its ''emittance''. In this case a simple formula still applies, namely
{{NumBlk2||<math display="block">A_\text{ellipse} = \pi\; y_\text{int}\, x_\text{max} = \pi\; x_\text{int}\, y_\text{max}</math>|3}}

where <math>y_{\text{int}}</math>, <math>x_{\text{int}}</math> are intercepts and <math>x_{\text{max}}</math>, <math>y_{\text{max}}</math> are maximum values. It follows directly from [[Ellipse#Theorem of Apollonios on conjugate diameters|Apollonios's theorem]].