\contentsline {chapter}{\numberline {0}Preface}{xi}{chapter.0}
\contentsline {chapter}{\numberline {1}The Integral}{1}{chapter.1}
\contentsline {section}{\numberline {1.1}Area}{1}{section.1.1}
\contentsline {section}{\numberline {1.2}Some applications of area}{6}{section.1.2}
\contentsline {subsection}{\numberline {1.2.1}Total Accumulation as ``Area''}{8}{subsection.1.2.1}
\contentsline {subsection}{\numberline {1.2.2}Problems}{8}{subsection.1.2.2}
\contentsline {section}{\numberline {1.3}Sigma notation and Riemann sums}{10}{section.1.3}
\contentsline {subsection}{\numberline {1.3.1}Sums of areas of rectangles}{12}{subsection.1.3.1}
\contentsline {subsection}{\numberline {1.3.2}Area under a curve –– Riemann sums}{14}{subsection.1.3.2}
\contentsline {subsection}{\numberline {1.3.3}Two special Riemann sums: lower and upper sums}{19}{subsection.1.3.3}
\contentsline {subsection}{\numberline {1.3.4}Problems}{20}{subsection.1.3.4}
\contentsline {subsection}{\numberline {1.3.5}The trapazoid rule}{21}{subsection.1.3.5}
\contentsline {subsection}{\numberline {1.3.6}Simpson's rule}{25}{subsection.1.3.6}
\contentsline {subsection}{\numberline {1.3.7}Trapazoidal vs. Simpson: Which Method Is Best?}{28}{subsection.1.3.7}
\contentsline {section}{\numberline {1.4}The definite integral}{29}{section.1.4}
\contentsline {subsection}{\numberline {1.4.1}The Fundamental Theorem of Calculus}{31}{subsection.1.4.1}
\contentsline {subsection}{\numberline {1.4.2}Problems}{34}{subsection.1.4.2}
\contentsline {subsection}{\numberline {1.4.3}Properties of the definite integral}{35}{subsection.1.4.3}
\contentsline {subsection}{\numberline {1.4.4}Problems}{37}{subsection.1.4.4}
\contentsline {section}{\numberline {1.5}Areas, integrals, and anti-derivatives}{39}{section.1.5}
\contentsline {subsection}{\numberline {1.5.1}Integrals, Antiderivatives, and Applications}{41}{subsection.1.5.1}
\contentsline {section}{\numberline {1.6}Indefinite Integrals and Change}{42}{section.1.6}
\contentsline {subsection}{\numberline {1.6.1}Indefinite Integrals}{42}{subsection.1.6.1}
\contentsline {subsection}{\numberline {1.6.2}Physical Intuition}{44}{subsection.1.6.2}
\contentsline {section}{\numberline {1.7}Substitution and Symmetry}{45}{section.1.7}
\contentsline {subsection}{\numberline {1.7.1}The Substitution Rule}{46}{subsection.1.7.1}
\contentsline {subsection}{\numberline {1.7.2}Changing the variable and definite integrals}{48}{subsection.1.7.2}
\contentsline {subsection}{\numberline {1.7.3}Symmetry}{50}{subsection.1.7.3}
\contentsline {subsection}{\numberline {1.7.4}Problems}{50}{subsection.1.7.4}
\contentsline {chapter}{\numberline {2}Applications}{53}{chapter.2}
\contentsline {section}{\numberline {2.1}Applications of the integral to area}{53}{section.2.1}
\contentsline {subsection}{\numberline {2.1.1}Using integration to determine areas}{53}{subsection.2.1.1}
\contentsline {section}{\numberline {2.2}Computing Volumes of Surfaces of Revolution}{58}{section.2.2}
\contentsline {subsection}{\numberline {2.2.1}Disc method}{60}{subsection.2.2.1}
\contentsline {subsection}{\numberline {2.2.2}Shell method}{63}{subsection.2.2.2}
\contentsline {section}{\numberline {2.3}Average Values}{65}{section.2.3}
\contentsline {section}{\numberline {2.4}Moments and centers of mass}{67}{section.2.4}
\contentsline {subsection}{\numberline {2.4.1}Point Masses}{67}{subsection.2.4.1}
\contentsline {subsection}{\numberline {2.4.2}Center of mass of a region in the plane}{69}{subsection.2.4.2}
\contentsline {subsection}{\numberline {2.4.3}$\overline {x}$ For A Region}{70}{subsection.2.4.3}
\contentsline {subsection}{\numberline {2.4.4}$\overline {y}$ For A Region}{71}{subsection.2.4.4}
\contentsline {subsection}{\numberline {2.4.5}Theorems of Pappus}{73}{subsection.2.4.5}
\contentsline {section}{\numberline {2.5}Arclengths}{74}{section.2.5}
\contentsline {subsection}{\numberline {2.5.1}2--d Arclength}{74}{subsection.2.5.1}
\contentsline {subsection}{\numberline {2.5.2}3--d Arclength}{77}{subsection.2.5.2}
\contentsline {chapter}{\numberline {3}Polar coordinates and complex numbers}{79}{chapter.3}
\contentsline {section}{\numberline {3.1}Polar Coordinates}{81}{section.3.1}
\contentsline {section}{\numberline {3.2}Areas in Polar Coordinates}{84}{section.3.2}
\contentsline {section}{\numberline {3.3}Complex Numbers}{87}{section.3.3}
\contentsline {subsection}{\numberline {3.3.1}Polar Form}{88}{subsection.3.3.1}
\contentsline {section}{\numberline {3.4}Complex Exponentials and Trig Identities}{91}{section.3.4}
\contentsline {subsection}{\numberline {3.4.1}Trigonometry and Complex Exponentials}{93}{subsection.3.4.1}
\contentsline {section}{\numberline {3.5}Integrals of Trigonometric Functions}{94}{section.3.5}