HMMT 二月 2006 · CALC 赛 · 第 6 题

英文解析

6. A triangle with vertices at (1003 , 0), (1004 , 3), and (1005 , 1) in the xy -plane is revolvedall the way around the y -axis. Find the volume of the solid thus obtained.
   Answer: 5020 π
   Solution: Let T ⊂ R denote the triangle, including its interior. Then T ’s area is 2
   5 / 2, and its centroid is (1004 , 4 / 3), so
   ∫
   x dx dy = · 1004 = 2510 .5
   ( x,y ) ∈ T2
   We are interested in the volume
   ∫
   2 πx dx dy,
   ( x,y ) ∈ Tbut this is just 2 π · 2510 = 5020 π .