A region is enclosed by the curves x2=4y, x2=−4y, x=4 and x=−4. V1 is the volume of the solid obtained by rotating the above region round the y-axis. Another region consists of points (x,y) satisfying x2+y2≤16, x2+(y−2)2≥4 and x2+(y+2)2≥4. V2 is the volume of the solid obtained by rotating this region round the y-axis. Then:
Pick one
Solution
As shown in the diagram, two solids of rotation obtained by rotating respectively two regions round the y-axis lie between two parallel planes, which are 8 units apart. We cut two solids of rotation by any plane which is perpendicular to the y-axis. Suppose the distance from the plane to the origin is ∣y∣≤4. Then the two sectional areas are
S1=π(42−4∣y∣), and S2=π(42−y2)−π[4−(2−∣y∣)2]=π(42−4∣y∣). So S1=S2. From the Zugen Principle (or Cavalieri Principle), we know that the volumes of the two geometric solids are equal, and that is, V1=V2. Answer: C.
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