Find the exact length of the curve. x = 9cos t − cos 9t, y = 9sin t − sin 9t, 0 ≤ t ≤ π?

Find the exact length of the curve.

x = 9cos t − cos 9t, y = 9sin t − sin 9t, 0 ≤ t ≤ π

Please show the steps! Thank you

2 Answers

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  • KevinM
    Lv 7
    9 years ago
    Favorite Answer

    The length will be:

    L = ∫ dL = ∫ sqrt(dx^2 + dy^2) = ∫ sqrt(1 + (dy/dx)^2) dx

    L = ∫ sqrt(1 + [(dy/dt) / (dx/dt)]^2) dx

    L = ∫ sqrt(1 + [(dy/dt) / (dx/dt)]^2) (dx/dt) dt

    L = ∫ sqrt((dx/dt)^2 + (dy/dt)^2) dt

    dx/dt = -9 sin t + 9 sin 9t

    dy/dt = 9 cos t - 9 cos 9t

    L = ∫ sqrt([9 sin 9t - 9 sin t]^2 + [9 cos t - 9 cos 9t]^2) dt

    = 9 ∫ sqrt( sin^2 9t - 2 sin 9t sin t + sin^2 t + cos^2 t - 2 cos 9t cos t + cos^2 9t) dt

    = 9 ∫ sqrt(2 - 2 sin 9t sin t - 2 cos 9t cos t) dt

    = 9 ∫ sqrt(2 - 2 (cos 8t - cos 10t)/2 - 2 (cos 8t + cos 10t)/2) dt

    = 9 ∫ sqrt(2 - 2 cos 8t) dt

    = 9 ∫ sqrt(2 - 2 (1 - 2 sin^2 4t) ) dt

    = 9 ∫ sqrt(2 - 2 + 4 sin^2 4t ) dt

    = 9 ∫ 2 |sin 4t| dt

    = -18/4 (4 * (cos 4t | t=0 to Pi/4) ) {The absolute value is important!}

    = -9/2 (4 * (-1 - 1) )= 36

    The exact length of this curve is 36 units.

  • Anonymous
    9 years ago

    its hard

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