5 | Integration | 291 | link | |
5.1 | Sums and Sigma Notation | 291 | link | |
Evaluating Sums | 293 | link | ||
5.2 | Areas as Limits of Sums | 296 | link | |
The Basic Area Problem | 297 | link | ||
Some Area Calculations | 298 | link | ||
5.3 | 5.3.1 Määrätty integraali | 302 | kirja | link |
5.3.2 Jaot ja Riemannin summat | 302 | kirja | link | |
5.3.3 Määrätty integraali | 303 | kirja | link | |
5.3.4 Yleiset Riemannin summat | 305 | kirja | link | |
5.3.5 Yleiset Riemannin summat, esimerkkejä | 305 | kirja | link | |
5.4 | Properties of the Definite Integral | 307 | link | |
A Mean-Value Theorem for Integrals | 310 | link | ||
Definite Integrals of Piecewise Continuous Functions | 311 | link | ||
5.5 | The Fundamental Theorem of Calculus | 313 | link | |
5.6 | The Method of Substitution | 319 | link | |
Trigonometric Integrals | 323 | link | ||
5.7 | Areas of Plane Regions | 327 | link | |
Areas Between Two Curves | 328 | link | ||
Chapter Review | 331 | link | ||
6 | Techniques of Integration | 334 | link |
6.1 | Integration by Parts | 334 | link |
Reduction Formulas | 338 | link | |
6.2 | Integrals of Rational Functions | 340 | link |
Linear and Quadratic Denominators | 341 | link | |
Partial Fractions | 343 | link | |
Completing the Square | 345 | link | |
Denominators with Repeated Factors | 346 | link | |
6.3 | Inverse Substitutions | 349 | link |
The Inverse Trigonometric Substitutions | 349 | link | |
Inverse Hyperbolic Substitutions | 352 | link | |
Other Inverse Substitutions | 353 | link | |
The tan(x/2) Substitution | 354 | link | |
6.4 | Other Methods for Evaluating Integrals | 356 | link |
The Method of Undetermined Coefficients | 357 | link | |
Using Maple for Integration | 359 | link | |
Using Integral Tables | 360 | link | |
Special Functions Arising from Integrals | 361 | link | |
6.5 | Improper Integrals | 363 | link |
Improper Integrals of Type I | 363 | link | |
Improper Integrals of Type II | 365 | link | |
Estimating Convergence and Divergence | 368 | link | |
6.6 | The Trapezoid and Midpoint Rules | 371 | link |
The Trapezoid Rule | 372 | link | |
The Midpoint Rule | 374 | link | |
Error Estimates | 375 | link | |
6.7 | Simpson's Rule | 378 | link |
6.8 | Other Aspects of Approximate Integration | 382 | link |
Approximating Improper Integrals | 383 | link | |
Using Taylor's Formula | 383 | link | |
Romberg Integration | 384 | link | |
The Importance of Higher-Order Methods | 387 | link | |
Other Methods | 388 | link | |
Chapter Review | 389 | link |
7 | Applications of Integration | 393 | link |
7.1 | Volumes by Slicing, Solids of Revolution | 393 | link |
Volumes by Slicing | 394 | link | |
Solids of Revolution | 395 | link | |
Cylindrical Shells | 398 | link | |
7.2 | More Volumes by Slicing | 402 | link |
7.3 | Arc Length and Surface Area | 406 | link |
Arc Length | 406 | link | |
The Arc Length of the Graph of a Function | 407 | link | |
Areas of Surfaces of Revolution | 410 | link | |
7.4 | Mass, Moments, and Centre of Mass | 413 | link |
Mass and Density | 413 | link | |
Moments and Centres of Mass | 416 | link | |
Two- and Three-Dimensional Examples | 417 | link | |
7.5 | Centroids | 420 | link |
Pappus's Theorem | 423 | link | |
7.6 | Other Physical Applications | 425 | link |
Hydrostatic Pressure | 426 | link | |
Work | 427 | link | |
Potential Energy and Kinetic Energy | 430 | link | |
7.7 | Applications in Business, Finance, and Ecology | 432 | link |
The Present Value of a Stream of Payments | 433 | link | |
The Economics of Exploiting Renewable Resources | 433 | link | |
7.8 | Probability | 436 | link |
Discrete Random Variables | 437 | link | |
Expectation, Mean, Variance, and Standard Deviation | 438 | link | |
Continuous Random Variables | 440 | link | |
The Normal Distribution | 444 | link | |
Heavy Tails | 447 | link | |
7.9 | First-Order Differential Equations | 450 | link |
Separable Equations | 450 | link | |
First-Order Linear Equations | 454 | link | |
Chapter Review | 458 | link |