HMMT 二月 2006 · 冲刺赛 · 第 29 题

HMMT February 2006 — Guts Round — Problem 29

[10] Find the area in the first quadrant bounded by the hyperbola x − y = 1, the x -axis, and the line 3 x = 4 y .

解析

Find the area in the first quadrant bounded by the hyperbola x − y = 1, the x -axis, and the line 3 x = 4 y . ln 7 Answer: 4 Solution: Convert to polar coordinates: the hyperbola becomes 2 2 2 2 1 = r (cos θ − sin θ ) = r cos(2 θ ) , so, letting α := arctan(3 / 4), the area is ∣ ∫ ∫ α α α 2 ∣ r 1 1 ∣ S := dθ = sec(2 θ ) dθ = ln | sec(2 θ ) + tan(2 θ ) | . ∣ 2 2 4 0 0 0 Now 2 tan α 3 / 2 24 tan(2 α ) = = = , 2 1 − tan α 7 / 16 7 √ 25 2 sec(2 α ) = 1 + tan (2 α ) = , 7 so ∣ ∣ ( ) ∣ ∣ 1 25 24 ln 7 ∣ ∣ S = ln + − ln | 1 + 0 | = . ∣ ∣ 4 7 7 4