China Mathematical Competition

Question

A region is enclosed by the curves x^2 = 4y, x^2 = -4y, x = 4 and x = -4. V_1 is the volume of the solid obtained by rotating the above region round the y-axis. Another region consists of points (x, y) satisfying x^2 + y^2 16, x^2 + (y-2)^2 4 and x^2 + (y+2)^2 4. V_2 is the volume of the solid obtained by rotating this region round the y-axis. Then ( ). (A) V_1 = 12V_2 (B) V_1 = 23V_2 (C) V_1 = V_2 (D) V_1 = 2V_2 FIGURE_0 FIGURE_1

Accepted Answer

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 ∣ \leq 4

. Then the two sectional areas are

S_{1} = π (4^{2} - 4∣ y ∣),

and

S_{2} = π (4^{2} - y^{2}) - π [4 - (2 - ∣ y ∣)^{2}] = π (4^{2} - 4∣ y ∣) .

So

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,

V_{1} = V_{2}

. Answer: C.