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Today's calculation of Integral 755

Source: 2011 Osaka University entrance exam

September 13, 2011
calculusintegrationtrigonometrygeometrygeometric transformationrotationcalculus computations

Problem Statement

Given mobile points P(0, sin⁡θ), Q(8cos⁡θ, 0) (0≤θ≤π2)P(0,\ \sin \theta),\ Q(8\cos \theta,\ 0)\ \left(0\leq \theta \leq \frac{\pi}{2}\right) on the xx-yy plane. Denote by DD the part in which line segment PQPQ sweeps. Find the volume VV generated by a rotation of DD around the xx-axis.
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