MathDB

Let

L

be a positive constant. For a point

P(t,\ 0)

on the positive part of the

x

axis on the coordinate plane, denote

Q(u(t),\ v(t))

the point at which the point reach starting from

P

proceeds by　distance

L

in counter-clockwise on the perimeter of a circle passing the point

P

with center

O

.
(1) Find

u(t),\ v(t)

.
(2) For real number

a

with

0<a<1

, find

f(a)=\int_a^1 \sqrt{\{u'(t)\}^2+\{v'(t)\}^2}\ dt

.
(3) Find

\lim_{a\rightarrow +0} \frac{f(a)}{\ln a}

.
2011 Tokyo University entrance exam/Science, Problem 3

686