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Today's calculation of Integral 636

Source:

August 14, 2010

calculusintegrationlogarithmsgeometrylimitderivativecalculus computations

Problem Statement

Let

a>1

be a constant. In the

xy

-plane, let

A(a,\ 0),\ B(a,\ \ln a)

and

C

be the intersection point of the curve

y=\ln x

and the

x

-axis. Denote by

S_1

the area of the part bounded by the

x

-axis, the segment

BA

and the curve

y=\ln x

(1) For

1\leq b\leq a

, let

D(b,\ \ln b)

. Find the value of

b

such that the area of quadrilateral

ABDC

is the closest to

S_1

and find the area

S_2

.
(2) Find

\lim_{a\rightarrow \infty} \frac{S_2}{S_1}

.
1992 Tokyo University entrance exam/Science

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