HMMT 二月 2005 · CALC 赛 · 第 6 题
HMMT February 2005 — CALC Round — Problem 6
题目详情
- The graph of r = 2 + cos 2 θ and its reflection over the line y = x bound five regions in the plane. Find the area of the region containing the origin.
解析
- The graph of r = 2 + cos 2 θ and its reflection over the line y = x bound five regions in the 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 is the 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 ) dθ π π 2 4 4 ( ) ∫ 3 π 1 4 = 2 4 + 4 cos 2 θ + (1 + cos 4 θ ) dθ π 2 4 3 π [ ] 4 1 = 9 θ + 4 sin 2 θ + sin 4 θ π 4 4 ( ) ( ) 27 π 9 π 9 π = − 4 − + 4 = − 8 . 4 4 2 2