Arc Length Formula for y = f(x): Setup, Simplification, and Domain Checks

The arc length of y = f(x) is ∫√(1 + (f′)²) dx, built from the hypotenuse of a small step along the curve. Most exercises are designed so the radicand collapses into a square; the absolute value that appears when the square root is taken is where mistakes happen.

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Find the Length of the Curve Over the Given Interval: y = (1/3)(x² − 2)^(3/2) on [2, 4]