HMMT 二月 2005 · CALC 赛 · 第 6 题

英文解析

6. The graph of r = 2 + cos 2 θ and its reflection over the line y = x bound five regions inthe plane. Find the area of the region containing the origin.
   Solution: 9 π/ 2 − 8
   π 3 π 5 π 7 π
   The original graph is closer to the origin than its reflection for θ ∈ ( , ) ∪ ( , ),
   4 4 4 4
   and the region is symmetric about the origin. Therefore the area we wish to find isthe polar integral
   3 π 3 π
   ∫ ∫
   4 1 4
   2 2
   4 (2 + cos 2 θ ) dθ = 2 (4 + 4 cos 2 θ + cos 2 θ ) dθπ π
   4 42
   ( )
   ∫ 3 π = 2 4 + 4 cos 2 θ + (1 + cos 4 θ ) dθ41
   2π
   3 π4
   [ ] = 9 θ + 4 sin 2 θ + sin 4 θ14
   4π
   ( ) ( )4
   27 π 9 π 9 π = − 4 − + 4 = − 8 .
   4 4 2 2