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Today's calculation of Integral 874

Source: 2013 Hitotsubashi University entrance exam, Problem 3

March 19, 2013
calculusintegrationconicsparabolageometrycalculus computations

Problem Statement

Given a parabola C:y=1−x2C : y=1-x^2 in xyxy-palne with the origin OO. Take two points P(p, 1−p2), Q(q, 1−q2) (p<q)P(p,\ 1-p^2),\ Q(q,\ 1-q^2)\ (p<q) on CC.
(1) Express the area SS of the part enclosed by two segments OP, OQOP,\ OQ and the parabalola CC in terms of p, qp,\ q.
(2) If q=p+1q=p+1, then find the minimum value of SS.
(3) If pq=−1pq=-1, then find the minimum value of SS.
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