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Compute the Arc Length of a Curve

Goldman SachsData Scientist · Online Assessment
EasyData ScientistOnline AssessmentStatistics & Math

Last updated

Tests single-variable calculus fundamentals, specifically calculating the derivative, setting up the standard arc length integral, and evaluating it using substitution.

Scenario

In quantitative finance and trajectory modeling, measuring the physical distance traversed along a smooth curve—such as a yield curve fitting spline or an asset pricing trajectory—requires computing curve arc lengths.

Consider the function:

y = (2/3) * x^(3/2) + 1

Your task is to compute the exact arc length of this curve along the interval [0, 1].

Instructions

Provide a full analytical derivation that covers the following three steps:

  1. First Derivative: Compute dy/dx and simplify the squared derivative (dy/dx)^2.
  2. Integral Formulation: State the standard arc length formula for a function y = f(x) on [a, b], and substitute dy/dx into the integrand sqrt(1 + (dy/dx)^2).
  3. Evaluation: Compute the definite integral from x = 0 to x = 1. Show the antiderivative and simplify your final exact numerical expression.

Requirements

  • Do not use numerical approximations (such as decimal approximations for square roots); express the final answer in exact form.
  • Clearly show intermediate steps for substitution and integration bounds.

Hints

Hint 1

Differentiating x^(3/2) produces a coefficient of 3/2, which cleanly cancels with the existing constant 2/3.

Hint 2

Notice that the expression inside the square root simplifies to 1 + x, avoiding the need for complex trigonometric substitutions.

Hint 3

Use the standard power rule to integrate (1 + x)^(1/2), noting that the antiderivative is (2/3) * (1 + x)^(3/2).

Follow-up questions

  • How does the arc length formula generalize when a curve is defined parametrically as x(t) and y(t)?
  • How would you numerically approximate the arc length if the integrand does not have an elementary antiderivative?
  • Why does the arc length integral guarantee that the computed length is always strictly greater than or equal to the straight-line Euclidean distance between the endpoints?