The circle above is centered at the origin and inscribed in the two curves and . Find the area of the red region above.
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Let P : ( x , 1 + x 2 1 ) and O : ( 0 , 0 )
To minimize O P :
D = ( O P ) 2 = x 2 + ( 1 + x 2 ) − 2 ⟹ d x d D = 2 x ( ( 1 + x 2 ) 3 ( 1 + x 2 ) 3 − 2 ) = 0
x = 0 ⟹ x = ± 2 3 1 − 1 ⟹ y = 2 3 1 1 ⟹ O P = 2 3 1 3 − 2 3 2
Let d = O P
I = ∫ − 2 3 1 − 1 2 3 1 − 1 ( x 2 + 1 1 − d 2 − x 2 ) d x
Let x = d sin ( θ ) ⟹ d x = d cos ( θ ) ⟹
I = ( 2 arctan ( x ) − d 2 arcsin ( d x ) + x d 2 − x 2 ) ∣ − 2 3 1 − 1 2 3 1 − 1 =
2 arctan ( 2 3 1 − 1 ) − ( 2 3 2 3 − 2 3 2 ) arcsin ( 3 − 2 3 2 2 3 1 2 3 1 − 1 ) − 2 3 1 2 3 1 − 1
⟹ The desired area A = 2 I ≈ 0 . 0 6 0 4 1 6 9 3 7 7 1 .
Note:
0 < x < 2 3 1 − 1 ⟹ d x d D < 0 and x > 2 3 1 − 1 ⟹ d x d D > 0
⟹ relative min at x = 2 3 1 − 1