3 Integral Calculus of Functions of one Variable
Integration reverses differentiation, and it also computes accumulated quantities such as area, volume and length. That these are the same operation is the content of the Fundamental Theorem of Calculus, and it is the reason the subject holds together.
The first part of the section is technique. Unlike differentiation, integration has no procedure guaranteed to succeed, so what exists instead is a collection of methods — substitution, parts, partial fractions, trigonometric and hyperbolic substitution — together with the judgement of which to try. That judgement is built by working examples, and the examples here are chosen to make the signals visible.
The second part applies the definite integral: areas between curves, volumes of revolution, arc length and surface area. In each case the pattern is the same and worth noticing in its own right — approximate the quantity by a sum of simple pieces, then pass to the limit — because it is the pattern by which integrals are set up in every later application.
3.2 Integration Formulas
3.2.1 Integration by Substitution
3.2.2 Integration by Parts
3.2.3 Reduction Formula
3.2.4 The Method of Undetermined Coefficient
3.3 Trigonometric Integrals
3.3.1 Method of Trigonometric Substitution
3.4 The Hyperbolic Substitution
3.5 Integration by Partial Fractions
3.6 Radicals of Polynomial Expression
3.7 Rational Functions of Trigonometric Functions
3.8 The Definite Integrals
3.9 The Fundamental Theorem of Calculus
3.10 Applications of Definite Integrals
3.10.1 Area Under the Curve
3.10.2 Area Between Two Curves
3.10.3 Volume of the Solid of Revolution
3.11 The Length of a Plane Curve
3.12 Area of Surface of Revolution
3.13 Practice Problems
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