MathDB

On the

x

y

plane, 3 half-lines

y=0,\ (x\geq 0),\ y=x\tan \theta \ (x\geq 0),\ y=-\sqrt{3}x\ (x\leq 0)

intersect with the circle with the center the origin

O

, radius

r\geq 1

A,\ B,\ C

respectively. Note that

\frac{\pi}{6}\leq \theta \leq \frac{\pi}{3}

.
If the area of quadrilateral

OABC

is one third of the area of the regular hexagon which inscribed in a circle with radius 1, then evaluate

\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} r^2d\theta .

2011 Waseda University of Education entrance exam/Science

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