Find The Length Of The Curve. R(T) = 4T, T2, 1 6 T3 , 0 ≤ T ≤ 1

Find The Length Of The Curve. R(T) = 4T, T2, 1 6 T3 , 0 ≤ T ≤ 1

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    Finding the Length of a Curve

    Calculating the length of a curve can be useful in a variety of circumstances. In this article, we will use the equation r(t) = 4t, t2, and 1 – 6t3 to find the length of the curve for 0 ≤ t ≤ 1.

    Steps to Find the Length of a Curve

    • Choose the equation that describes the curve.
    • Integrate the equation.
    • Substitute the limits of integration.
    • Calculate the length of the curve.

    Calculating the Length of the Curve

    The equation we are using to find the length of the curve is r(t) = 4t, t2, and 1 – 6t3 for 0 ≤ t ≤ 1.

    We can find the length of the curve by integrating the equation and then substituting the limits of integration. The integral of the equation is:

    L = ∫01√[(4t)2 + (t2)2 + (1 – 6t3)2]dt

    We can now calculate the length of the curve by substituting the limits of integration:

    L = ∫01√[(4t)2 + (t2)2 + (1 – 6t3)2]dt = 2.12

    Therefore, the length of the curve with the equation r(t) = 4t, t2, and 1 – 6t3 for 0 ≤ t ≤ 1 is 2.12.

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