MathDB

Today's calculation of Integral 570

Source: 2010 Tokyo Institute of Technology entrance exam, Problem 1, 70pts

February 27, 2010

calculusintegrationtrigonometryfunctioncalculus computations

Problem Statement

Let f(x) \equal{} 1 \minus{} \cos x \minus{} x\sin x. (1) Show that f(x) \equal{} 0 has a unique solution in

0 < x < \pi

. (2) Let J \equal{} \int_0^{\pi} |f(x)|dx. Denote by

\alpha

the solution in (1), express

J

in terms of

\sin \alpha

. (3) Compare the size of

J

defined in (2) with

\sqrt {2}