CONTENTS 3.2.1 What is Area? 3.2.3 Riemann Sum 183 3.3 Numerical Calculation of Integral 3.3.1 Left and right Rule 3.3.2 Midpoint Rule and Trapezoidal Rule 186 3.3.3 Simpson's rule 189 3.4 Indefinite Integral 3.4.1 Fundamental Theorem of Calculus 3.4.2 Indefinite Integral 3.5 Properties of Integration 198 3.5.1 Linear Property 198 3.5.2 Integration by Part 204 3.5.3 Change of Variable 214 3.6 Integration of Rational Function 228 3.6.1 Rational function r+b 3.6.2 Rational Function of 3.6.3 Rational Function of sin a and cos x 3.7 Improper Integral 240 3.7.1 Definition and Property 240 3.7.2 Comparison Test 245 3.7.3 Conditional Convergence 249 3.8 Application to Geometry 3.8.1 Length of Curve 253 3.8.2 Area of Region 3. 8.3 Surface of Revolution 3.8.4 Solid of Revolution 5 Cavalieri's Principle 3.9 Polar Coordinate 3.9.1 Curves in Polar Coordinate 3.9.2 Geometry in Polar Coordinate 3.10 Application to Physics 3.10.1 Work and Pressure 3.10.2 Center of Mass 4 Series 297 4.1 Series of numbers 4.1.1 Sum of se erles 298 4. 1.2 Property of Converging Series 300 4.2 Comparison Test 4.2.1 Integral test 4.2.2 Comparison TestCONTENTS 5 3.2.1 What is Area? . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 3.2.2 Darboux Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 3.2.3 Riemann Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 3.3 Numerical Calculation of Integral . . . . . . . . . . . . . . . . . . . . 184 3.3.1 Left and Right Rule . . . . . . . . . . . . . . . . . . . . . . . 184 3.3.2 Midpoint Rule and Trapezoidal Rule . . . . . . . . . . . . . . 186 3.3.3 Simpson’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 189 3.4 Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 3.4.1 Fundamental Theorem of Calculus . . . . . . . . . . . . . . . 191 3.4.2 Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . 195 3.5 Properties of Integration . . . . . . . . . . . . . . . . . . . . . . . . . 198 3.5.1 Linear Property . . . . . . . . . . . . . . . . . . . . . . . . . . 198 3.5.2 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . 204 3.5.3 Change of Variable . . . . . . . . . . . . . . . . . . . . . . . . 214 3.6 Integration of Rational Function . . . . . . . . . . . . . . . . . . . . . 228 3.6.1 Rational Function . . . . . . . . . . . . . . . . . . . . . . . . . 228 3.6.2 Rational Function of n r ax + b cx + d . . . . . . . . . . . . . . . . . . 235 3.6.3 Rational Function of sin x and cos x . . . . . . . . . . . . . . . 238 3.7 Improper Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 3.7.1 Definition and Property . . . . . . . . . . . . . . . . . . . . . 240 3.7.2 Comparison Test . . . . . . . . . . . . . . . . . . . . . . . . . 245 3.7.3 Conditional Convergence . . . . . . . . . . . . . . . . . . . . . 249 3.8 Application to Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 253 3.8.1 Length of Curve . . . . . . . . . . . . . . . . . . . . . . . . . . 253 3.8.2 Area of Region . . . . . . . . . . . . . . . . . . . . . . . . . . 258 3.8.3 Surface of Revolution . . . . . . . . . . . . . . . . . . . . . . . 265 3.8.4 Solid of Revolution . . . . . . . . . . . . . . . . . . . . . . . . 269 3.8.5 Cavalieri’s Principle . . . . . . . . . . . . . . . . . . . . . . . . 276 3.9 Polar Coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 3.9.1 Curves in Polar Coordinate . . . . . . . . . . . . . . . . . . . 284 3.9.2 Geometry in Polar Coordinate . . . . . . . . . . . . . . . . . . 288 3.10 Application to Physics . . . . . . . . . . . . . . . . . . . . . . . . . . 292 3.10.1 Work and Pressure . . . . . . . . . . . . . . . . . . . . . . . . 292 3.10.2 Center of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . 295 4 Series 297 4.1 Series of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 4.1.1 Sum of Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 4.1.2 Property of Converging Series . . . . . . . . . . . . . . . . . . 300 4.2 Comparison Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 4.2.1 Integral Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 4.2.2 Comparison Test . . . . . . . . . . . . . . . . . . . . . . . . . 305