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Today's calculation of Integral 778

Source: 2012 Nippon Medical School entrance exam

February 3, 2012
calculusintegrationgeometry3D geometryspherecalculus computations

Problem Statement

In the xyzxyz space with the origin OO, Let K1K_1 be the surface and inner part of the sphere centered on the point (1, 0, 0)(1,\ 0,\ 0) with radius 2 and let K2K_2 be the surface and inner part of the sphere centered on the point (−1, 0, 0)(-1,\ 0,\ 0) with radius 2. For three points P, Q, RP,\ Q,\ R in the space, consider points X, YX,\ Y defined by OX→=OP→+OQ→, OY→=13(OP→+OQ→+OR→).\overrightarrow{OX}=\overrightarrow{OP}+\overrightarrow{OQ},\ \overrightarrow{OY}=\frac 13(\overrightarrow{OP}+\overrightarrow{OQ}+\overrightarrow{OR}).
(1) When P, QP,\ Q move every cranny in K1, K2K_1,\ K_2 respectively, find the volume of the solid generated by the whole points of the point XX.
(2) Find the volume of the solid generated by the whole points of the point RR for which for any PP belonging to K1K_1 and any QQ belonging to K2K_2, YY belongs to K1K_1.
(3) Find the volume of the solid generated by the whole points of the point RR for which for any PP belonging to K1K_1 and any QQ belonging to K2K_2, YY belongs to K1∪K2K_1\cup K_2.
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