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HMMT 二月 2006 · CALC 赛 · 第 6 题

HMMT February 2006 — CALC Round — Problem 6

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. A triangle with vertices at (1003 , 0), (1004 , 3), and (1005 , 1) in the xy -plane is revolved all the way around the y -axis. Find the volume of the solid thus obtained.
解析
  1. A triangle with vertices at (1003 , 0), (1004 , 3), and (1005 , 1) in the xy -plane is revolved all the way around the y -axis. Find the volume of the solid thus obtained. Answer: 5020 π 2 Solution: Let T ⊂ R denote the triangle, including its interior. Then T ’s area is 5 / 2, and its centroid is (1004 , 4 / 3), so ∫ 5 x dx dy = · 1004 = 2510 . 2 ( x,y ) ∈ T We are interested in the volume ∫ 2 πx dx dy, ( x,y ) ∈ T but this is just 2 π · 2510 = 5020 π .