HMMT 二月 2009 · CALC 赛 · 第 9 题

HMMT February 2009 — CALC Round — Problem 9

[ 7 ] Let R be the region in the plane bounded by the graphs of y = x and y = x . Compute the volume of the region formed by revolving R around the line y = x .

解析

[ 7 ] let R be the region in the plane bounded by the graphs of y = x and y = x . Compute the volume of the region formed by revolving R around the line y = x . √ 2 π Answer: 60 Solution: We integrate from 0 to 1 using the method of washers. Fix d between 0 and 1. Let the 2 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 R √ 2 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 the expression [( x − x ) / 2] 2 dx , and the answer follows.