\contentsline {chapter}{\numberline {1}Preface}{5}
\contentsline {section}{\numberline {1.1}Computers}{5}
\contentsline {chapter}{\numberline {2}Definite and Indefinite Integrals}{7}
\contentsline {section}{\numberline {2.1}The Definite Integral}{7}
\contentsline {subsection}{\numberline {2.1.1}The definition of area under curve}{7}
\contentsline {subsection}{\numberline {2.1.2}Relation between velocity and area}{7}
\contentsline {subsection}{\numberline {2.1.3}Definition of Integral}{8}
\contentsline {subsection}{\numberline {2.1.4}The Fundamental Theorem of Calculus}{8}
\contentsline {section}{\numberline {2.2}Indefinite Integrals and Change}{9}
\contentsline {subsection}{\numberline {2.2.1}Indefinite Integrals}{9}
\contentsline {subsection}{\numberline {2.2.2}Examples}{10}
\contentsline {subsection}{\numberline {2.2.3}Physical Intuition}{11}
\contentsline {section}{\numberline {2.3}Substitution and Symmetry}{12}
\contentsline {subsection}{\numberline {2.3.1}The Substitution Rule}{12}
\contentsline {subsection}{\numberline {2.3.2}The Substitution Rule for Definite Integrals}{14}
\contentsline {subsection}{\numberline {2.3.3}Symmetry}{14}
\contentsline {chapter}{\numberline {3}Applications to Areas, Volume, and Averages}{17}
\contentsline {section}{\numberline {3.1}Using Integration to Determine Areas Between Curves}{17}
\contentsline {subsection}{\numberline {3.1.1}Examples}{18}
\contentsline {section}{\numberline {3.2}Computing Volumes of Surfaces of Revolution}{20}
\contentsline {section}{\numberline {3.3}Average Values}{22}
\contentsline {chapter}{\numberline {4}Polar Coordinates and Complex Numbers}{25}
\contentsline {section}{\numberline {4.1}Polar Coordinates}{25}
\contentsline {section}{\numberline {4.2}Areas in Polar Coordinates}{27}
\contentsline {subsection}{\numberline {4.2.1}Examples}{28}
\contentsline {section}{\numberline {4.3}Complex Numbers}{29}
\contentsline {subsection}{\numberline {4.3.1}Polar Form}{30}
\contentsline {section}{\numberline {4.4}Complex Exponentials and Trig Identities}{32}
\contentsline {subsection}{\numberline {4.4.1}Trigonometry and Complex Exponentials}{34}
\contentsline {chapter}{\numberline {5}Integration Techniques}{37}
\contentsline {section}{\numberline {5.1}Integration By Parts}{37}
\contentsline {section}{\numberline {5.2}Trigonometric Integrals}{40}
\contentsline {subsection}{\numberline {5.2.1}Some Remarks on Using Complex-Valued Functions}{43}
\contentsline {section}{\numberline {5.3}Trigonometric Substitutions}{44}
\contentsline {section}{\numberline {5.4}Factoring Polynomials}{48}
\contentsline {section}{\numberline {5.5}Integration of Rational Functions Using Partial Fractions}{49}
\contentsline {section}{\numberline {5.6}Approximating Integrals}{53}
\contentsline {section}{\numberline {5.7}Improper Integrals}{56}
\contentsline {subsection}{\numberline {5.7.1}Convergence, Divergence, and Comparison}{59}
\contentsline {chapter}{\numberline {6}Sequences and Series}{63}
\contentsline {section}{\numberline {6.1}Sequences}{63}
\contentsline {section}{\numberline {6.2}Series}{64}
\contentsline {section}{\numberline {6.3}The Integral and Comparison Tests}{66}
\contentsline {subsection}{\numberline {6.3.1}Estimating the Sum of a Series}{70}
\contentsline {section}{\numberline {6.4}Tests for Convergence}{71}
\contentsline {subsection}{\numberline {6.4.1}The Comparison Test}{71}
\contentsline {subsection}{\numberline {6.4.2}Absolute and Conditional Convergence}{72}
\contentsline {subsection}{\numberline {6.4.3}The Ratio Test}{72}
\contentsline {subsection}{\numberline {6.4.4}The Root Test}{73}
\contentsline {section}{\numberline {6.5}Power Series}{75}
\contentsline {subsection}{\numberline {6.5.1}Shift the Origin}{76}
\contentsline {subsection}{\numberline {6.5.2}Convergence of Power Series}{76}