HMMT 二月 2006 · 冲刺赛 · 第 29 题

英文解析

29. 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:
    Solution: Convert to polar coordinates: the hyperbola becomes 4
    2 2 2 2
    1 = r (cos θ − sin θ ) = r cos(2 θ ) ,
    so, letting α := arctan(3 / 4), the area is
    ∣
    ∫ ∫
    α αα
    ∣2
    r 1 1
    ∣
    S := dθ = sec(2 θ ) dθ = ln | sec(2 θ ) + tan(2 θ ) | .
    ∣
    2 2 4
    0 0
    Now 0
    2 tan α 3 / 2 24
    tan(2 α ) = = = ,
    1 − tan α 7 / 16 72
    √
    sec(2 α ) = 1 + tan (2 α ) = ,225
    so 7
    ∣ ∣
    ( )
    ∣ ∣
    1 25 24 ln 7
    ∣ ∣
    S = ln + − ln | 1 + 0 | = .
    ∣ ∣
    4 7 7 4