Mathematics

Find the area bounded by the ellipse $$\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$$ and the ordinates $$x=0$$ and $$x=ae$$, where $$b^{2}=a^{2}(1-e^{2})$$ and $$e<1$$


SOLUTION
Required area $$A$$ is given by 
$$A=2$$ (Area of shaded region in first quadrant)
$$\Rightarrow A=2 \displaystyle\int_{0}^{ae}|y|\ dx=2\displaystyle\int_{0}^{ae}y\ dx$$                      $$[\because y\ge 0\therefore |y|=y]$$
$$\Rightarrow A=2\dfrac{b}{a}\left[\displaystyle\int_{0}^{ae}\sqrt{a^{2}-x^{2}}dx\right]$$
$$\Rightarrow A=\dfrac{2b}{a}\left[\dfrac{1}{2}\times \sqrt{a^{2}-x^{2}}+\dfrac{1}{2}a^{2}\sin^{-1}\dfrac{x}{a}\right]_{ae}^{0}$$
$$\Rightarrow A=\dfrac{2b}{a}\left[\dfrac{ae}{2}\sqrt{a^{2}-a^{2}e^{2}}+\dfrac{1}{2}a^{2}\sin^{-1}e\right]$$
$$\Rightarrow A=\dfrac{b}{a}\left(a^{2}e\sqrt{1-e^{2}}+a^{2}\sin^{-1}e\right)=ab\left(e\sqrt{1-e^{2}}+\sin^{-1}e\right)$$
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Subjective Medium Published on 17th 09, 2020
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