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Today's calculation of Integral 749

Source: 2011 Gakusyuin University entrance exam/Psychology

August 31, 2011
calculusintegrationconicsparabolageometrycalculus computations

Problem Statement

Let mm be a positive integer. A tangent line at the point PP on the parabola C1:y=x2+m2C_1 : y=x^2+m^2 intersects with the parabola C2:y=x2C_2 : y=x^2 at the points A, BA,\ B. For the point QQ between AA and BB on C2C_2, denote by SS the sum of the areas of the region bounded by the line AQAQ,C2C_2 and the region bounded by the line QBQB, C2C_2. When QQ move between AA and BB on C2C_2, prove that the minimum value of SS doesn't depend on how we would take PP, then find the value in terms of mm.
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