Integral Calculus
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This text develops infinite series and integration with mathematical precision, building directly on the foundations laid in Differential Calculus. Just as sequences in Calculus 1 provided the intuition for limits and continuity, series extend this to infinite sums, revealing the power of convergence in modeling accumulation. Linear maps and differentials from Calculus 1 clarify integration as the inverse of differentiation, transforming global problems into local ones.
Contents
Infinite Series: Definition via partial sums, geometric and telescoping series, convergence tests (comparison, ratio, root, alternating series, integral test)
Power Series: Radius of convergence, term-by-term differentiation and integration, operations on power series
Taylor Series: Taylor’s theorem with remainder, standard expansions, applications to approximation and computation
The Riemann Integral: Riemann sums, definition, integrability theorems, properties of integrals
The Fundamental Theorem of Calculus: Both parts, connection between differentiation and integration
Integration Techniques: Substitution, integration by parts, trigonometric integrals, partial fractions
Applications of Integration: Area, volume, arc length, work, center of mass
Improper Integrals: Infinite intervals, discontinuous integrands, convergence
Why Series First?
Most calculus sequences introduce integration before series. We reverse this order.
Since we’ve already developed sequence convergence in differential calculus, series follow naturally: an infinite series is simply a sequence of partial sums. We can immediately define convergence, prove tests, and develop power series theory—all building on familiar foundations.
Taylor series emerge as extensions of linear approximation (polynomials approximating functions). When we reach integration, we can use power series as a tool: integrating term-by-term, evaluating difficult integrals through series expansions.
Prerequisites
Differential calculus including sequences and their convergence, limits, continuity, and derivatives. Familiarity with the material in Differential Calculus or equivalent.