HMMT 二月 2004 · CALC 赛 · 第 7 题
HMMT February 2004 — CALC Round — Problem 7
题目详情
- Find the area of the region in the xy -plane satisfying x − x + y ≤ 0. 4
解析
- 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 calculate the 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