Integral Calculus - NU

1. The Definite and Indefinite IntegralAntiderivatives: Finding general formulas for functions whose derivatives are known.Riemann Sums: Approximating areas under curves using finite geometric rectangular summations.The Fundamental Theorem of Calculus (FTC): Connecting differentiation and integration through Parts 1 and 2 of the theorem.2. Advanced Integration TechniquesIntegration by Substitution: Shifting variables using u-substitution to simplify compound functions.Integration by Parts: Evaluating products of functions using the classical \(\int u \, dv = uv - \int v \, du\) structure.Trigonometric Integrals & Substitutions: Managing products of trigonometric powers and eliminating radical forms using \(x = a\sin\theta\), \(x = a\tan\theta\), or \(x = a\sec\theta\).Partial Fractions: Decomposing complex rational polynomial expressions into simpler, integrable fractions.3. Geometric and Physical ApplicationsArea Between Curves: Setting up integrals to calculate the exact area enclosed by multiple intersecting functions.Volumes of Solids of Revolution: Computing volumes using the Disk Method, Washer Method, and Cylindrical Shell Method.Arc Length & Surface Area: Using integration to compute the structural length of curves and areas of surfaces of revolution.4. Improper Integrals and Improper BoundsInfinite Intervals: Evaluating integrals where one or both of the integration boundaries extend to infinity (±∞).Discontinuous Integrands: Integrating functions that possess vertical asymptotes or infinite discontinuities within the interval of integration.