HMMT 二月 2010 · 冲刺赛 · 第 20 题

英文解析

20. [ 10 ] Find the volume of the set of points ( x, y, z ) satisfyingx, y, z ≥ 0
    x + y ≤ 1
    y + z ≤ 1
    z + x ≤ 1
    Answer: Without loss of generality, assume that x ≥ y - half the volume of the solid is on this 1
    side of the plane x = y . For each value of c from 0 to , the region of the intersection of this half of 14
    the solid with the plane y = c is a trapezoid. The trapezoid has height 1 − 2 c and average base , so 12
    it has an area of − c .12
    1 1 1 12
    The total volume of this region is times the average area of the trapezoids, which is · = . Double
    2 2 4 8
    that to get the total volume, which is .1 4