How do you find the arc length function for the curve defined by the equation y = 2x^(3/2), starting at the point P(0, 9) and ending at P(4, 54)?

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To find the arc length function for the curve defined by the equation y = 2x^(3/2), we will follow these steps:

  1. Understand the Arc Length Formula: The arc length L of a curve y = f(x) from x = a to x = b is given by:

    2. L = ∫ from a to b √(1 + (dy/dx)²) dx

  2. Find the Derivative: Calculate the derivative of the function to find dy/dx.

    3. The curve given is y = 2x^(3/2). To find the derivative:

    dy/dx = d/dx (2x^(3/2)) = 3x^(1/2).

  3. Substitute into the Arc Length Formula: Now replace dy/dx in the arc length formula.

    4. We have:

    L = ∫ from a to b √(1 + (3x^(1/2))²) dx

    This simplifies to:

    L = ∫ from 0 to 4 √(1 + 9x) dx

  4. Evaluate the Integral: We will need to integrate √(1 + 9x).

    5. To solve this integral, we can perform a substitution. Let:

    u = 1 + 9x, thus du = 9 dx ⇒ dx = du/9.

    Next, change the limits of integration: when x = 0, u = 1 and when x = 4, u = 37.

    Then the integral becomes:

    L = ∫ from 1 to 37 (√u) (du/9) = (1/9) ∫ from 1 to 37 u^(1/2) du.

    The integral of u^(1/2) is:

    (2/3) u^(3/2).

  5. Calculating the Result: Substitute back in:

    6. L = (1/9) * (2/3) [u^(3/2)] from 1 to 37.

    This gives:

    L = (2/27) [37^(3/2) – 1^(3/2)]

    Calculating 37^(3/2) yields approximately 226.765. So:

    L ≈ (2/27) * (226.765 – 1) ≈ (2/27) * 225.765.

  6. Conclusion: The arc length from the point P(0, 9) to P(4, 54) along the curve y = 2x^(3/2) is approximately:

    7. L ≈ 16.717.

This gives you the arc length function along the specified curve!

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