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Class-12-science»Maths

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c u r v e x^{2} + y^{2} \leq 2 a x (x - a)^{2} + y^{2} \leq a^{2} r e p r e s e n t s a r e a i n s i d e t h e c i r c l e w i t h c e n t r e (a, 0) a n d r a d i u s a y^{2} \geq a x r e p r e s e n t s a r e a o u t s i d e t h e p a r a b o l a w i t h f o c u s (a / 4, 0) t w o c u r v e s i n t e r s e c t a t x^{2} + y^{2} = 2 a x x^{2} + a x = 2 a x x^{2} - a x = 0 x = 0 o r x = a p o i n t s a r e (0, 0) a n d (a, a) r e q u i r e d a r e a = a r e a i n f i r s t q u a d r a n t (s h a d e d a r e a i n f i g u r e) = \int_{0}^{a} (\sqrt{2 a x - x^{2}} - \sqrt{a x}) d x = \int_{0}^{a} (\sqrt{a^{2} - (x - a)^{2}} - \sqrt{a x}) d x = {[\frac{1}{2} \{- (a - x) \sqrt{a^{2} - (x - a)^{2}} + a^{2} \sin^{- 1} \frac{a - x}{a}\} - \frac{2 \sqrt{a} x^{\frac{3}{3}}}{3}]}^{a}_{0} = \frac{1}{2} \{- (a - a) \sqrt{a^{2} - (a - a)^{2}} + a^{2} \sin^{- 1} \frac{a - a}{a}\} - \frac{2 \sqrt{a} a^{\frac{3}{3}}}{3} - \{- \frac{1}{2} \{- (a - 0) \sqrt{a^{2} - (0 - a)^{2}} + a^{2} \sin^{- 1} \frac{a - 0}{a}\} - \frac{2 \sqrt{a} 0^{\frac{3}{3}}}{3}\} = \frac{1}{2} a^{2} \sin^{- 1} \frac{a}{a} - \frac{2}{3} a^{2} = \frac{π}{2} a^{2} - \frac{2}{3} a^{2} = (\frac{3 π - 4}{6}) a^{2}

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