Determine the Arc Length of a Curve

For the curve y = (2/3)x^(3/2) + 1 on the interval 0 <= x <= 1, work out its arc length. Present the derivative, the integral that gives arc length, and the exact result.

Constraints & Assumptions

• Apply the standard Euclidean arc-length formula for a differentiable graph.
• On this interval, the fractional power is taken to mean the nonnegative real value.

Clarifying Questions to Ask

• Would an exact expression alone be enough, or is a decimal approximation also expected?
• Do you want the setup obtained 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

• The derivative computed correctly on the interval.
• The right substitution into sqrt(1 + (dy/dx)^2).
• The definite integral evaluated accurately.
• A brief sanity check against the straight-line distance.

Follow-up Questions

• What happens to the result on 0 <= x <= a?
• Why can arc length never be smaller than the distance between the endpoints?