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Geometry Difficulty 4.4 AIME Find the answer China

A region is enclosed by the curves x2=4yx^2 = 4y, x2=−4yx^2 = -4y, x=4x = 4 and x=−4x = -4. V1V_1 is the volume of the solid obtained by rotating the above region round the y-axis. Another region consists of points (x,y)(x, y) satisfying x2+y2≤16x^2 + y^2 \le 16, x2+(y−2)2≥4x^2 + (y-2)^2 \ge 4 and x2+(y+2)2≥4x^2 + (y+2)^2 \ge 4. V2V_2 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|y| \le 4. Then the two sectional areas are

S1=π(42−4∣y∣), S_1 = \pi(4^2 - 4|y|),
and
S2=π(42−y2)−π[4−(2−∣y∣)2]=π(42−4∣y∣). S_2 = \pi(4^2 - y^2) - \pi[4 - (2 - |y|)^2] = \pi(4^2 - 4|y|).
So
S1=S2.S_1 = S_2.
From the Zugen Principle (or Cavalieri Principle), we know that the volumes of the two geometric solids are equal, and that is, V1=V2V_1 = V_2. Answer: C.

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