Let O be the origin. For α 0 , f ( x ) = 1 + x 2 1 , A = ( α , f ( α )) , B = ( − 1/ α , f ( − 1/ α )) . Find the minimum area bounded by OA, OB and the curve’s arc from B to A.

Question

Let O be the origin. For α > 0 , f ( x ) = 1 + x 2 1 ​ , A = ( α , f ( α )) , B = ( − 1/ α , f ( − 1/ α )) . Find the minimum area bounded by OA, OB and the curve’s arc from B to A. ​

TestHub · Accepted Answer

Subtract the two triangles under OA and OB from the area under the curve: S = ∫ − 1/ α α ​ 1 + x 2 d x ​ − 2 α f ( α ) ​ − 2 α f ( − 1/ α ) ​ = 2 π ​ − 1 + α 2 α ​ . 1 + α 2 α ​ ≤ 2 1 ​ S m i n ​ = 2 π − 1 ​ ( α = 1 ) . ​

Mathematics - Area Under Curves Question with Solution | TestHub

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