Arc Length of a Curve
Calculate the arc length of curves using integration and the distance formula. Derive and apply the arc length formula L = ∫[a to b] √(1 + [f'(x)]²)dx for functions y = f(x), or L = ∫[α to β] √([dx/dt]² + [dy/dt]²)dt for parametric curves. Understand how the Pythagorean theorem leads to this formula by summing infinitesimal line segments along the curve.
Integration by Parts
Master integration by parts, derived from the product rule for differentiation, using the formula ∫u dv = uv - ∫v du. Learn strategic selection of u and dv using the LIATE priority (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential). Practice solving integrals involving products of functions, handling repeated applications, and recognizing when tabular integration or reduction formulas provide efficient solutions.