find-the-common-area-of-x-2-3-y-2-1-x-2-y-2-3-1-

Question Number 57821 by behi83417@gmail.com last updated on 12/Apr/19

\boldsymbol find \boldsymbol the \boldsymbol common \boldsymbol area \boldsymbol of :

{\begin{cases} \frac{\boldsymbol x^{2}}{3} + \boldsymbol y^{2} = 1 \\ \boldsymbol x^{2} + \frac{\boldsymbol y^{2}}{3} = 1 \end{cases}

Answered by MJS last updated on 13/Apr/19

4 (\underset{0}{\int^{\frac{\sqrt{3}}{2}}} \sqrt{1 - \frac{x^{2}}{3}} d x + \underset{\frac{\sqrt{3}}{2}}{\int^{1}} \sqrt{3 - 3 x^{2}} d x) = \frac{2 \sqrt{3}}{3} π

Answered by mr W last updated on 13/Apr/19

r^{2} = \frac{a^{2} b^{2}}{b^{2} \cos^{2} θ + a^{2} \sin^{2} θ}

A_{c o m m o n} = 8 \int_{0}^{\frac{π}{4}} \frac{r^{2} d θ}{2} = 4 a^{2} b^{2} \int_{0}^{\frac{π}{4}} \frac{d θ}{b^{2} \cos^{2} θ + a^{2} \sin^{2} θ}

= 4 a b {[\tan^{- 1} (\frac{a}{b} \tan θ)]}_{0}^{\frac{π}{4}}

= 4 a b \tan^{- 1} (\frac{a}{b})

w i t h a = 1, b = \sqrt{3}

A_{c o m m o n} = 4 \sqrt{3} \tan^{- 1} (\frac{1}{\sqrt{3}}) = \frac{4 \sqrt{3} π}{6} = \frac{2 \sqrt{3} π}{3}

Commented by behi83417@gmail.com last updated on 13/Apr/19

t h a n k y o u s o m u c h s i r M J S a n d s i r M r W .

t h a n k y o u s o m u c h s i r M J S a n d s i r M r W .