HMMT 二月 2004 · CALC 赛 · 第 7 题

英文解析

7. Find the area of the region in the xy -plane satisfying x − x + y ≤ 0.
   Solution: π/ 2
   √
   2 6
   Rewrite the condition as | y | ≤ x − x . The right side is zero when x is − 1, 0, or 1,
   and it bounds an area symmetric about the x - and y -axes. Therefore, we can calculatethe area by the integral
   ∫ ∫ ∫
   1 1 1
   √ √ √
   2 6 4 2
   2 x − x dx = 4 x 1 − x dx = 2 1 − u du = π/ 2 .
   − 1 0 0 4