📂Geometry

Calculating the Area of an Ellipse Using Integration

Formula

The area of the ellipse $\displaystyle {x^2 \over a^2} + {y^2 \over b^2} = 1$ is $ab \pi$.

Explanation

Especially $a=b=r$, that is, the area of a circle $x^2 + y^2=r^2$ with radius $r$ is as well known $r^2 \pi$.

Proof

To obtain the area of an ellipse, it is sufficient to calculate only the area of the shaded region. The area of the region is given by $$\int _{0} ^{a} \sqrt{b^2-{b^2 \over a^2} x^2} dx$$ By substituting $x = a \sin \theta$, we get $$\begin{align*} \int _{0} ^{ \pi \over 2 } b \sqrt{1 - \sin ^ 2 \theta } a \cos \theta d \theta =& ab \int _{0} ^{ \pi \over 2 } \cos ^2 \theta d \theta \\ =& ab \left[ {1 \over 4} (2\theta + \sin 2\theta)\right]_{0}^{\pi \over 2} \\ =& {ab \over 4} \pi \end{align*}$$ Multiplying this by $4$ yields the area of the ellipse $ab \pi$.

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See Also

• Circumference of an ellipse