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Find the area of the region bounded by the curves y 2 = 2 x and x 2 + y 2 = 4 x .

0 2 Views | Posted 3 months ago
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    Answered by

    Payal Gupta | Contributor-Level 10

    3 months ago

    This is a long answer type question as classified in NCERT Exemplar

    E q u a t i o n s     o f     t h e     c u r v e     a r e     g i v e n     b y                                                             x 2 + y 2 = 4 x                                                                                                     … ( i ) a n d                                                                   y 2 = 2 x                                                                                                   … ( i i ) ⇒                                 x 2 − 4 x + y 2 = 0 ⇒ x 2 − 4 x + 4 − 4 + y 2 = 0                                           ( x − 2 ) 2 + y 2 = 4 C l e a r l y     i t     i s     t h e     e q u a t i o n     o f     a     c i r c l e     h a v i n g     i t s     c e n t r e ( 2 , 0 )     a n d     r a d i u s     2 . S o l v i n g     x 2 + y 2 = 4 x     a n d     y 2 = 2 x                                     x 2 + 2 x = 4 x ⇒       x 2 + 2 x − 4 x = 0 ⇒                             x 2 − 2 x = 0 ⇒                         x ( x − 2 ) = 0 ∴                                                             x = 0 , 2 Area  of  the  required  region=2[∫024−(x−2)2dx−∫022xdx]       [ ?     P a r a b o l a     a n d     c i r c l e     b o t h     a r e     s y m m e t r i c a l     a b o u t     x − a x i s ]                                                                   = 2 [ x − 2 2 4 − ( x − 2 ) 2 + 4 2 s i n − 1 x − 2 2 ] 0 2 − 2 . 2 . 2 3 [ x 3 / 2 ] 0 2                                                                   = 2 [ ( 0 + 0 ) − ( 0 + 2 s i n − 1 ( − 1 ) ) ] −

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