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636

Part of 2010 Today's Calculation Of Integral

Problems(1)

Today's calculation of Integral 636

Source:

8/14/2010
Let a>1a>1 be a constant. In the xyxy-plane, let A(a, 0), B(a, ln⁡a)A(a,\ 0),\ B(a,\ \ln a) and CC be the intersection point of the curve y=ln⁡xy=\ln x and the xx-axis. Denote by S1S_1 the area of the part bounded by the xx-axis, the segment BABA and the curve y=ln⁡xy=\ln x
(1) For 1≤b≤a1\leq b\leq a, let D(b, ln⁡b)D(b,\ \ln b). Find the value of bb such that the area of quadrilateral ABDCABDC is the closest to S1S_1 and find the area S2S_2.
(2) Find lim⁡a→∞S2S1\lim_{a\rightarrow \infty} \frac{S_2}{S_1}.
1992 Tokyo University entrance exam/Science
calculusintegrationlogarithmsgeometrylimitderivativecalculus computations