# Compute the Arc Length of a Curve Find the arc length of `y = (2/3)x^(3/2) + 1` over `0 <= x <= 1`. Show the derivative, the arc-length integral, and the exact final value. ### Constraints & Assumptions - Use the ordinary Euclidean arc-length formula for a differentiable graph. - The fractional power denotes the nonnegative real value on this interval. ### Clarifying Questions to Ask - Is an exact expression sufficient, or is a decimal approximation also wanted? - Should the setup be derived from the general arc-length formula? ```hint Differentiate before simplifying The coefficient is chosen so the square of the derivative makes the integrand simple. ``` ### What a Strong Answer Covers - Correct derivative on the interval. - Correct substitution into `sqrt(1 + (dy/dx)^2)`. - Accurate evaluation of the definite integral. - A quick plausibility check against the straight-line distance. ### Follow-up Questions - How would the result change over `0 <= x <= a`? - Why must arc length be at least the distance between the endpoints?