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Math Help - Arc length

  1. #1
    Junior Member
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    Arc length

    Find the length of the curve.
    t^2i + 2tj + lntk, 1 <=t <=e
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by purplerain View Post
    Find the length of the curve.
    t^2i + 2tj + lntk, 1 <=t <=e
    Arc length: \int_1^e\sqrt{(dx/dt)^2+(dy/dt)^2+(dz/dt)^2}\,dt

    \frac{dx}{dt}=2t
    \frac{dy}{dt}=2
    \frac{dz}{dt}=\frac{1}{t}

    Spoiler:
    \int_1^e\sqrt{4t^2+4+\frac{1}{t^2}}\,dt = \int_1^e\sqrt{\frac{4t^4+4t^2+1}{t^2}}\,dt =\int_1^e\frac{\sqrt{(2t^2+1)^2}}{t}\,dt =\int_1^e\frac{2t^2+1}{t}\,dt=\int_1^e 2t+\frac{1}{t}\,dt = \left[t^2+\ln t\right]_1^e = e^2+\ln e - 1^2-\ln 1 =e^2+1-1-0= \boxed{e^2}.
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