How to Find the Arc Length Function for the Curve y = 2x^3 with Starting Point P(0, 54)

To find the arc length function for the curve defined by the equation y = 2x^3, we use the arc length formula:

L = ∫√(1 + (dy/dx)²) dx

First, we need to calculate the derivative dy/dx. For the given curve:

y = 2x³

Taking the derivative:

dy/dx = 6x²

Now, we substitute this into our arc length formula:

We have:

1 + (dy/dx)² = 1 + (6x²)²

This simplifies to:

1 + 36x^4

Now we can plug this back into the integral:

L = ∫√(1 + 36x⁴) dx

Next, we can compute this integral from the starting point P(0, 54) to a variable endpoint x:

L(x) = ∫₀^x √(1 + 36t⁴) dt

This function will give us the arc length from the starting point (0, 54) to any point (x, y) on the curve. To completely evaluate the arc length, you would perform the integration, which might be best done using numerical methods or software, as it does not yield a simple antiderivative.