Calculus 1 — 30.2: Arc Length: Perfect Squares & U-Substitution скачать в хорошем качестве

Calculus 1 — 30.2: Arc Length: Perfect Squares & U-Substitution

Two fully worked arc length examples reveal the key algebra strategies that make these integrals tractable. Learn to recognize when the integrand collapses into a perfect square trinomial and when u-substitution is the right tool — plus how choosing between x and y parameterizations can dramatically simplify your work. Key concepts covered: • The arc length formula L = ∫√(1+[f'(x)]²) dx and its three-stage workflow: set up, simplify, evaluate • Perfect square trinomial recognition: spotting the sign flip from −1/2 to +1/2 after adding 1 • Using the (a−b)² and (a+b)² expansion patterns to factor complex expressions under the radical • Temporary variable substitution (letting u = x⁴) to reveal hidden trinomial structure • U-substitution for arc length when a linear expression appears under the square root • Changing integration bounds directly into u so no back-substitution is needed • Comparing x-parameterization vs y-parameterization on the same curve (y = x^(3/2)) • How the y-approach yields integer bounds (13 to 22) while the x-approach gives fractional bounds • Simplifying fractional exponents like (11/2)^(3/2) and (13/4)^(3/2) • Strategic planning: checking which derivative and which bounds are cleaner before committing to a parameterization