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Today's calculation of Integral 817

Source: 2012 Osaka University entrance exam/Science, 2nd exam

May 16, 2012
calculusintegrationfunctionlogarithmsgeometrycalculus computations

Problem Statement

Define two functions f(t)=12(t+1t), g(t)=t2−2ln⁡tf(t)=\frac 12\left(t+\frac{1}{t}\right),\ g(t)=t^2-2\ln t. When real number tt moves in the range of t>0t>0, denote by CC the curve by which the point (f(t), g(t))(f(t),\ g(t)) draws on the xyxy-plane. Let a>1a>1, find the area of the part bounded by the line x=12(a+1a)x=\frac 12\left(a+\frac{1}{a}\right) and the curve CC.
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