Question

Find the area bounded by the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and ordinates $x=0$ and $x=ae$.

Answer 1

The equation of ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

or $\frac{y^2}{b^2}=1-\frac{x^2}{a^2}$

or $y^2=b^2(1-\frac{x^2}{a^2})$

or $y^2=\frac{b^2}{a^2}(a^2-x^2)$

or $y=\frac{b}{a}\sqrt{a^2-x^2}$ since it is first quadrant.

Ordiantes are $x=0,x=ae$

Required area$=2{\int }_0^{ae}ydx$ since the ellipse is symmetrical about $x-$ axis.

$=\frac{2b}{a}{\int }_0^{ae}\sqrt{a^2-x^2}dx$ .....$\left(1\right)$

$=\frac{2b}{a}{[\frac{x\sqrt{a^2-x^2}}{2}+\frac{a^2}{2}{\sin }^{-1}\frac{x}{a}]}_0^{ae}$

$=\frac{2b}{a}[\{\frac{ae\sqrt{a^2-a^2{e}^2}}{2}+\frac{a^2}{2}{\sin }^{-1}\left(\frac{x}{a}\right)\}-\{0-\frac{a^2}{2}{\sin }^{-1}0\}]$

$=\frac{b}{a}[ae\times a\sqrt{1-e^2}+a^2{\sin }^{-1}e]$

$=ab[e\sqrt{1-e^2}+{\sin }^{-1}e]$