How can I find the exact length of the curve defined by the equation y = 5 – 8x^3 + 2x + 1?

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Finding the Length of the Curve y = 5 – 8x³ + 2x + 1

To find the exact length of a curve defined by a function, you can use the formula for arc length. The general formula for the length of the curve from
x = a to x = b is given by:

        L = ∫_a^b &sqrt;(1 + (dy/dx)²) dx

In our case, we have:

        y = 5 - 8x³ + 2x + 1

First, we need to find
dy/dx:

        dy/dx = d/dx (5 - 8x³ + 2x + 1) = -24x² + 2

Now, we substitute this expression into the arc length formula. The next step is to calculate the square of the derivative:

        (dy/dx)² = (-24x² + 2)² = 576x^4 - 96x² + 4

Now we add 1:

        1 + (dy/dx)² = 1 + 576x^4 - 96x² + 4 = 576x^4 - 96x² + 5

The length of the curve from
x = a to x = b is now:

        L = ∫_a^b &sqrt;(576x⁴ - 96x² + 5) dx

Calculating this integral gives you the exact length of the curve over the specified range. In many cases, this integral may not have a closed-form solution and may need to be computed using numerical methods or graphing software.

If you have specific values for a and b, we can further narrow down the calculation. For instance, if you plan to find the length from
x = 0 to x = 1, substitute those limits into the integral. In this case:

        L = ∫_0^1 &sqrt;(576x⁴ - 96x² + 5) dx

Lastly, evaluating this integral via numerical methods—or using software tools like MATLAB, Python, or even graphing calculators—will yield the required length for the curve.

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