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Today's calculation of Integral 868

Source: 2013 The University of Tokyo/Science, Problem 6

February 25, 2013
calculusintegrationanalytic geometryinequalitiesgeometrygeometric transformationrotation

Problem Statement

In the coordinate space, define a square SS, defined by the inequality ∣x∣≤1, ∣y∣≤1|x|\leq 1,\ |y|\leq 1 on the xyxy-plane, with four vertices A(−1, 1, 0), B(1, 1, 0), C(1,−1, 0),D(−1,−1, 0)A(-1,\ 1,\ 0),\ B(1,\ 1,\ 0),\ C(1,-1,\ 0), D(-1,-1,\ 0). Let V1V_1 be the solid by a rotation of the square SS about the line BDBD as the axis of rotation, and let V2V_2 be the solid by a rotation of the square SS about the line ACAC as the axis of rotation.
(1) For a real number tt such that 0≤t<10\leq t<1, find the area of cross section of V1V_1 cut by the plane x=tx=t.
(2) Find the volume of the common part of V1V_1 and V2V_2.
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