How do I find the arc length function for the curve defined by the equation y = 2x^3 with the starting point at P(0, 12)?

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To find the arc length function for the curve given by the equation y = 2x³ from the starting point P(0, 12), we will use the formula for arc length.

The arc length L from point a to point b is given by the integral:

L = ∫ab sqrt(1 + (dy/dx)²) dx

First, we need to find the derivative of y with respect to x:

y = 2x³
dy/dx = d(2x³)/dx = 6x²

Now we substitute the expression for dy/dx into the arc length formula:

L = ∫0x sqrt(1 + (6x²)²) dx

Next, simplify the expression inside the square root:

L = ∫0x sqrt(1 + 36x⁴) dx

Now, we need to evaluate this integral:

The arc length function will typically be expressed in terms of the variable x and can be challenging to compute in closed form. Nevertheless, the expression we have is:

L(x) = ∫0x sqrt(1 + 36x⁴) dx

To compute L(x), one may use numerical integration methods or special functions, depending on the desired precision and application. In most practical applications, numerical approximation methods—like Simpson’s Rule or the Trapezoidal Rule—can be employed.

Finally, recall that the arc length function gives the
total length of the curve between the starting point and any point x on the curve.

In summary, the arc length function for the curve y = 2x³ from the point P(0, 12) is represented by the integral:

L(x) = ∫0x sqrt(1 + 36x⁴) dx

This expresses the arc length from the starting point to any point along the curve as a function of x.

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