HMMT 二月 2009 · CALC 赛 · 第 9 题

英文解析

9. [ 7 ] let R be the region in the plane bounded by the graphs of y = x and y = x . Compute the volumeof the region formed by revolving R around the line y = x .
   √
   2 π
   Answer:
   Solution: We integrate from 0 to 1 using the method of washers. Fix d between 0 and 1. Let the 60
   line x = d intersect the graph of y = x at Q , and let the line x = d intersect the graph of y = x at 2
   P . Then P = ( d, d ), and Q = ( d, d ). Now drop a perpendicular from Q to the line y = x , and let R2
   √
   be the foot of this perpendicular. Because P QR is a 45 − 45 − 90 triangle, QR = ( d − d ) / 2. So the 2
   √ √
   differential washer has a radius of ( d − d ) / 2 and a height of 2 dx . So we integrate (from 0 to 1)2
   √ √
   2 2
   the expression [( x − x ) / 2] 2 dx , and the answer follows.