Calculus II

The goal of this text is to help students leam to use calculus intelligently for solving a wide variety of mathematical and physical problems. This book is an outgrowth of our teaching of calculus at Berkeley, and the present edition incorporates many improvements based on our use of the first edition. We list below some of the key features of the book. Examples and Exercises The exercise sets have been carefully constructed to be of maximum use to the students. With few exceptions we adhere to the following policies. - The section exercises are graded into three consecutive groups: (a) The first exercises are routine, modelIed almost exactly on the exam pIes; these are intended to give students confidence. (b) Next come exercises that are still based directly on the examples and text but which may have variations of wording or which combine different ideas; these are intended to train students to think for themselves. (c) The last exercises in each set are difficult. These are marked with a star (*) and some will challenge even the best students. Difficult does not necessarily mean theoretical; often a starred problem is an interesting application that requires insight into what calculus is really about. - The exercises come in groups of two and often four similar ones.

7 Basic Methods of Integration. - 7. 1 Calculating Integrals. - 7. 2 Integration by Substitution. - 7. 3 Changing Variables in the Definite Integral. - 7. 4 Integration by Parts. - 8 Differential Equations. - 8. 1 Oscillations. - 8. 2 Growth and Decay. - 8. 3 The Hyperbolic Functions. - 8. 4 The Inverse Hyperbolic Functions. - 8. 5 Separable Differential Equations. - 8. 6 Linear First-Order Equations. - 9 Applications of Integration. - 9. 1 Volumes by the Slice Method. - 9. 2 Volumes by the Shell Method. - 9. 3 Average Values and the Mean Value Theorem for Integrals. - 9. 4 Center of Mass. - 9. 5 Energy, Power, and Work. - 10 Further Techniques and Applications of Integration. - 10. 1 Trigonometric Integrals. - 10. 2 Partial Fractions. - 10. 3 Arc Length and Surface Area. - 10. 4 Parametric Curves. - 10. 5 Length and Area in Polar Coordinates. - 11 Limits, L Hôpital s Rule, and Numerical Methods. - 11. 1 Limits of Functions. - 11. 2 L Hôpital s Rule. - 11. 3 Improper Integrals. - 11. 4 Limits of Sequences and Newton s Method. - 11. 5 Numerical Integration. - 12 Infinite Series. - 12. 1 The Sum of an Infinite Series. - 12. 2 The Comparison Test and Alternating Series. - 12. 3 The Integral and Ratio Tests. - 12. 4 Power Series. - 12. 5 Taylor s Formula. - 12. 6 Complex Numbers. - 12. 7 Second-Order Linear Differential Equations. - 12. 8 Series Solutions of Differential Equations. - Answers.