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Geometry Volume

Source: 2002 National High School Mathematics League, Exam One, Problem 6

March 13, 2020
geometryrotation

Problem Statement

Consider the area encircled by x2=4y,x2=−4y,x=4,x=−4x^2=4y,x^2=-4y,x=4,x=-4, rotate it around yy-axis, the volume of the revolved body is V1V_1. Then consider the figure formed by all points (x,y)(x,y) that x2+y2≤16,x2+(y−2)2≥4,x2+(y−2)2≥4x^2+y^2\leq16,x^2+(y-2)^2\geq4,x^2+(y-2)^2\geq4, rotate it around yy-axis, the volume of the revolved body is V2V_2. The relationship between V1V_1 and V2V_2 is (A)V1=12V2(B)V1=23V2(C)V1=V2(D)V1=2V2\text{(A)}V_1=\frac{1}{2}V_2\qquad\text{(B)}V_1=\frac{2}{3}V_2\qquad\text{(C)}V_1=V_2\qquad\text{(D)}V_1=2V_2
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