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Calculus 2 Notes∗ Calculus 2 notes∗ Attila M´at´e Brooklyn College of the City University of New York July 27, 2020 Contents Contents 1 1 Inverse trigonometric functions 5 1.1 Theinverseofafunction ........................... ........ 5 1.2 Theinverseofsine................................ ....... 6 1.2.1 The derivative of arcsin x ............................. 7 1.3 Theinverseofcosine .............................. ....... 7 1.3.1 The derivative of arccos x ............................. 8 1.4 Theinverseoftangent ............................. ....... 9 1.4.1 The derivative of arctan x ............................. 10 1.5 Theinverseofsecant .............................. ....... 11 1.5.1 The derivative of arcsec x ............................. 12 1.6 Integrals leading to inverse trigonometric functions . ................... 12 1.7 Reading ......................................... 13 1.8 Homework ........................................ 13 2 Applications of the definite integral 14 2.1 TheRiemannIntegral.............................. ....... 14 2.2 Areabetweencurves ............................... ...... 15 2.3 Volumeofrotationwithslices . .......... 15 2.4 Volumeofrotationwithshells . .......... 16 2.5 Work............................................ 16 2.6 Arclength ....................................... 17 2.7 Reading ......................................... 18 2.8 Homework ........................................ 18 3 Integration by parts 19 3.1 Change of variables in indefinite integrals . ............... 19 3.2 Changeofvariablesindefiniteintegrals . .............. 19 3.3 Integration by parts for indefinite integrals . ................ 20 3.4 Integration by parts for definite integrals . ............... 20 ∗Written for the course Mathematics 1206 (Calculus 2) at Brooklyn College of CUNY. 1 3.5 Examples........................................ 21 3.6 Reading ......................................... 26 3.7 Homework ........................................ 26 4 Trigonometric integrals 26 4.1 Integrationofproductsofsineandcosine . .............. 26 4.2 Integration of products of tangent and secant . .............. 27 4.3 Product of sine and cosine of multiple angles . .............. 29 4.4 Reading ......................................... 30 4.5 Homework ........................................ 30 5 Evaluationofintegralswithsquarerootofquadratics 30 5.1 Trigonometricsubstitutions . ........... 30 5.1.1 Integrals with √a2 x2 .............................. 30 5.1.2 Integrals with √x2 −+ a2 .............................. 31 5.1.3 Integrals with √x2 a2 .............................. 31 5.2 Eulersubstitutions ..............................− ........ 31 5.3 FirstEulersubstitution. .......... 31 5.4 SecondEulersubstitution . ......... 32 5.5 ThirdEulersubstitution . ......... 32 5.6 Sometimestherearesimplerways . .......... 32 5.6.1 Aretheresultsreallydifferent? . ........ 34 6 Examples for trigonometiric substituition 36 6.1 Reading ......................................... 37 6.2 Homework ........................................ 37 7 Integration of rational functions: partial fraction decomposition 37 7.1 An example for form of partial fraction decomposition . ................ 38 7.2 Finding the coefficents with linear factors, no multiple zeros ............... 39 7.3 Finding the coefficents with linear factors, multiple zeros................. 40 7.4 Quadraticfactors ................................ ....... 41 8 Integration of rational functions 42 8.1 Integrating partial fractions with linear denominator ................... 42 8.2 Fractions with quadratic denominators: integrating the part with x in the numerator . 42 8.3 Fractions with quadratic denominators: integrating them with constant numerator . 43 9 Integration techniques: reducing the exponent of a quadratic denominator 43 9.1 A recursive formula for integrals with quadratic denominators .............. 43 9.2 An example: trigonometric substitutions for integrals with quadratic denominators . 44 9.3 An example: extending the idea of recursion with powers of quadratic denominators . 45 9.4 Reading ......................................... 46 9.5 Homework ........................................ 46 10 Moreonintegration:theWeierstrasssubstitution 46 10.1 Reading ........................................ 48 10.2 Homework ....................................... 48 2 11 The l’Hˆopital rule 48 11.1 Rolle’stheorem ................................. ....... 48 11.2 TheCauchyMean-ValueTheorem. ......... 49 11.3 Interchange of limits and continuos functions . ................. 51 11.4 The exponential function compared to the power function ................ 51 11.5 Thepowerfunctionandthelogarithm . ........... 52 11.6 Examples: l’Hˆopital’s rule is not just for fractions . .................... 52 11.7 The standard definition of the natural exponential function................ 53 11.8 Reading ........................................ 53 11.9 Homework ....................................... 53 12 Improper integrals 54 12.1 Thedefinitionofimproperintegrals . ............. 54 12.2 One needs to be careful with improper integrals . ............... 55 12.3 Integrals of powers of x oninfiniteintervals. 56 12.3.1 Between α = 1 and α< 1........................... 56 12.4 Comparison test for integrals− with a− limit at infinity . .................. 57 12.5 Limit comparison test for integrals with a limit at infinity ................ 57 12.6 Integralswithlowerlimitatzero. ............. 58 12.7 Comparison tests for integrals with lower limit at zero .................. 58 12.8 Reading ........................................ 59 12.9 Homework ....................................... 59 13 Polar coordinates 59 13.1 Representation of points in polar coordinates . ................. 59 13.2 Parametricdescriptionofcurves . ............. 60 13.3 Description of curves in polar coordinates. ................ 60 13.3.1 Thales’stheorem . 61 13.4 Setsandtheirgraphs ............................. ........ 61 13.5 Reading ........................................ 61 13.6 Homework ....................................... 61 14 Area and arclength in polar coordinates 62 14.1 Area ........................................... 62 14.2 Arclength ...................................... ..... 64 14.3 Intersection of curves in polar coordinates . ................ 65 14.4 Anexample ...................................... 65 14.5 Reading ........................................ 67 14.6 Homework ....................................... 67 15 The completeness of real numbers 68 15.1 Upperbound,supremum . ....... 68 15.2 Lowerbound,infimum .............................. ...... 69 16 Sequences and limits 70 16.1 Sequencesandsubsequences . .......... 70 16.2 Limits ......................................... 70 16.2.1 A subsequence of a convergent subsequence is convergent ........... 72 16.3 Limitrules..................................... ...... 72 3 16.4 Reading ........................................ 72 16.5 Homework ....................................... 72 17 Series 73 17.1 Thegeometricseries. ......... 73 17.2 Moretelescopingsums . ........ 74 17.3 Limitrulesforseries ............................ ......... 75 17.4 TheBaselproblem ................................ ...... 76 17.5 Examples....................................... ..... 76 17.6 Reading ........................................ 77 17.7 Homework ....................................... 77 18 The comparison test and the integral test 78 18.1 Thecomparisontest.............................. ........ 78 18.2 Theintegraltest................................ ........ 78 18.3 Estimating the partial sums and the tails of a series . ................. 79 18.4 Reading ........................................ 81 18.5 Homework ....................................... 81 19 The limit comparison test 81 19.1 Anexample ...................................... 82 19.2 Homework ....................................... 83 20 The alternating series test 83 20.1 Errorofthealternatingseries . ............ 85 20.2 Anexample ...................................... 85 20.3 TheDirichletconvergencetest . ............ 86 20.4 Reading ........................................ 86 20.5 Homework ....................................... 86 21 Absolute convergence; comparing to geometric series 86 21.1 Conditionally convergent series cannot be rearranged ................... 87 21.2 TheRatioTest ................................... ..... 87 21.2.1 TheLimitRatioTest . 89 21.3 TheRootTest.................................... ..... 90 21.3.1 TheLimitRootTest. 90 21.4 Reading ........................................ 91 21.5 Homework ....................................... 91 22 Miscellaneous convergence problems 91 22.1 Reading ........................................ 93 22.2 Homework ....................................... 93 23 Power series 94 23.1 Example for calculating the radius of convergence . ................. 94 23.2 Limit superior: the real truth about the radius of convergence .............. 96 23.2.1 Anexample.................................... 97 23.3 Powerseriesnotcenteredattheorigin . ............. 97 23.4 Reading ........................................ 98 4 23.5 Homework ....................................... 98 24 Representing functions with power series 98 24.1 Examples....................................... ..... 99 24.2 Reading ........................................ 100 24.3 Homework ....................................... 100 25 Taylor and Maclaurin series 100 25.1 A function not represented by its Maclaurin series . ................
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