匿名
匿名 發問於 科學及數學數學 · 4 年 前

arc length?

find the arc length of the curve y=x^(2/3) where -1≦x≦8

1 個解答

評分
  • Berm
    Lv 4
    4 年 前
    最佳解答

    y' = (2/3)x^(-1/3)

    (y')² = 4/(9x^2/3)

    ∫ [-1 to 8] (1 + (y')²)^(1/2) dx

    = ∫ [-1 to 8] (1 + 4/(9x^2/3))^(1/2) dx

    = ∫ [-1 to 8] ( (9x^(2/3)+4)/(9x^2/3) )^(1/2) dx

    = ∫ [-1 to 8] (1/(3x^(1/3))) * (9x^(2/3)+4)^(1/2) dx

    = ∫ [-1 to 8] (3/(3x^(1/3))) * (x^(2/3)+4/9)^(1/2) dx

    = ∫ [-1 to 8] (1/x^(1/3)) * (x^(2/3)+4/9)^(1/2) dx

    = (x^(2/3)+4/9)^(3/2) from x= -1 to 8

    we can split it to (-1,0) and (0,8)

    since y=x^(2/3) an even function

    (-1,0) = (0,1)

    That is,

    = (x^(2/3)+4/9)^(3/2) from x=0 to 8 + (x^(2/3)+4/9)^(3/2) from x=0 to 1

    = [(40/9)^(3/2) - 8/27] + [(13/9)^(3/2) - 8/27]

    = 9.07342 + 1.43971

    = 10.51313

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