LUTHER COLLEGE UNIVERSITY OF REGINA Math 111 Calculus II by Robert G. Petry and Fotini Labropulu published by Campion College and Luther College 2.5.2 Inverse Cosine . 31 2.5.3 Inverse Tangent . 32 2.5.4 Other Trigonometric Inverses . 33 2.6 L'H^opital'sRule . 35 2.6.1 Indeterminate Forms of Contents type 0 · 1 and 1 − 1 .. 36 2.6.2 Exponential Indeterminate Forms . 37 3 Integration Methods 39 1 Integration Review1 3.1 Integration by Parts . 40 1.1 The Meaning of the Definite 3.2 Trigonometric Integrals . 42 Integral . .2 3.3 Trigonometric Substitution . 45 1.2 The Fundamental Theorem of 3.4 Partial Fraction Decomposition . 47 Calculus . .3 3.5 General Strategies for Integration 51 1.3 Indefinite Integrals . .4 3.6 Improper Integrals . 52 1.4 Integration by Substitution . .5 3.6.1 Improper Integrals of the 1.5 Integration Examples . .6 First Kind . 52 3.6.2 Improper Integrals of the 2 Inverses and Other Functions7 Second Kind . 53 2.1 Inverse Functions . .8 2.1.1 Horizontal Line Test . .8 4 Sequences and Series 57 2.1.2 Finding Inverse Functions 10 4.1 Sequences . 58 2.1.3 Graphs of Inverse Functions 11 4.2 Series . 63 2.1.4 Derivative of an Inverse 4.3 Testing Series with Positive Terms 67 Function . 12 4.3.1 The Integral Test . 67 2.1.5 Creating Invertible 4.3.2 The Basic Comparison Test 70 Functions . 12 4.3.3 The Limit Comparison Test 71 2.2 Exponential Functions . 13 4.4 The Alternating Series Test . 73 2.2.1 The Natural Exponential 4.5 Tests of Absolute Convergence . 75 Function . 14 4.5.1 Absolute Convergence . 75 2.2.2 Derivative of ex ...... 16 4.5.2 The Ratio Test . 75 2.2.3 Integral of ex ....... 16 4.5.3 The Root Test . 76 2.2.4 Simplifying Exponential 4.5.4 Rearrangement of Series . 77 Expressions . 17 4.6 Procedure for Testing Series . 78 2.3 Logarithmic Functions . 18 4.7 Power Series . 80 2.3.1 Logarithmic Function 4.8 Representing Functions with Properties . 19 Power Series . 83 2.3.2 The Natural Logarithmic 4.9 Maclaurin Series . 85 Function . 20 4.10 Taylor Series . 87 2.3.3 Solving Exponential and Logarithmic Equations . 21 5 Integration Applications 91 2.3.4 Derivative of the Natural 5.1 Areas Between Curves . 92 Logarithmic Function . 22 5.2 Calculation of Volume . 94 2.3.5 Derivatives Using 5.2.1 Volume as a Calculus Arbitrary Bases . 23 Problem . 94 2.3.6 Logarithmic Differentiation 23 5.2.2 Solids of Revolution . 95 1 x 2.3.7 Integral of x and a ... 25 5.2.3 The Disk Method . 96 2.4 Exponential Growth and Decay . 26 5.2.4 The Washer Method . 98 2.5 Inverse Trigonometric Functions . 29 5.3 The Shell Method . 101 2.5.1 Inverse Sine . 29 5.4 Determining the Volume Method 105 * Sections denoted with an asterisk are optional and may be omitted in the course. 2 CONTENTS 3 5.5 Arc Length . 107 6.2.3 Arc Length . 118 5.6 Areas of Surfaces of Revolution . 110 7 Polar Coordinates 119 6 Parametric Equations 113 7.1 Polar Coordinates . 120 6.1 Parametric Equations . 114 7.1.1 Converting Between Polar 6.2 Calculus of Parametric Curves . 116 and Cartesian Coordinates 121 6.2.1 Tangent Slope and 7.1.2 Curves in Polar Coordinates 121 Concavity . 116 7.1.3 Symmetry in Polar Curves 122 6.2.2 Area Under the Curve . 117 7.1.4 Tangents . 122 4 Unit 1: Integration Review 1 2 1.1 The Meaning of the Definite Integral 1.1 The Meaning of the Definite Integral The definite integral of the function f(x) between x = a and x = b is written: Z b f(x) dx a Geometrically it equals the area A between the curve y = f(x) and the x-axis between the vertical lines x = a and x = b: y y = f(x) A x a b More precisely, assuming a < b, the definite integral is the net sum of the signed areas between the curve y = f(x) and the x-axis where areas below the x-axis (i.e. where f(x) dips below the x-axis) are counted negatively. R b R The notation used for the definite integral, a f(x) dx, is elegant and intuitive. We are umming (R dA) the (infinitesimally) small differential rectangular areas dA = f(x)·dx of height f(x) and width dx at each value x between x = a and x = b: y y = f(x) dx dA A f(x) x a x b We will see soon how viewing integrals as sums of differentials can be used to come up with formulas for calculations aside from just area. Integration Review 3 1.2 The Fundamental Theorem of Calculus As seen in a previous calculus course, the definite integral can be written as a limiting sum (Riemann Sum) of N rectangles of finite width ∆x = (b − a)=N where we let the number of rectangles (N) go to infinity (and consequently the width ∆x ! 0). This method of evaluating a definite integral is hard or impossible to compute exactly yfor most functions. An easy way to evaluate definite integrals is due to the Fundamental Theorem of Calculus which relates the calculation of a definite integral with the evaluation of the antiderivative F (x) of f(x): Theorem: 1.1. The Fundamental Theorem of Calculus: If f is continuous on [a; b] then Z b f(x) dx = F (b) − F (a) a for any F an antiderivative of f, i.e. F 0(x) = f(x). b Notationally we write F (b) − F (a) with the shorthand F (x)ja, i.e. b F (x)ja = F (b) − F (a) ; where, unlike the integral sign, the bar is placed on the right. 4 1.3 Indefinite Integrals 1.3 Indefinite Integrals Because of the intimate relationship between the antiderivative and the definite integral, we define the indefinite integral of f(x) (with no limits a or b) to just be the antiderivative, i.e. Z f(x) dx = F (x) + C where F (x) is an antiderivate of f(x) (so F 0(x) = f(x)) and C is an arbitrary constant. The latter is d required since the antiderivative of a function is not unique as dx C = 0 implies we can always add a constant to an antiderivative to get another antiderivative of the same function. Using our notation for indefinite integrals and our knowledge of derivatives gives the following. Table of Indefinite Integrals Z 1 1. xn dx = xn+1 + C (n 6= −1) n + 1 Z 2. cos x dx = sin x + C Z 3. sin x dx = − cos x + C Z 4. sec2 x dx = tan x + C Z 5. sec x tan x dx = sec x + C Z 6. csc2 x dx = − cot x + C Z 7. csc x cot x dx = − csc x + C Z Z 8. cf(x) dx = c f(x) dx Z Z Z 9. [f(x) ± g(x)] dx = f(x) dx ± g(x) dx In the last two integration formulae f(x) and g(x) are functions while c is a constant. For indefinite 1 n+1 n n integrals we say, for example, that n+1 x + C is the (indefinite) integral of x where x is the integrand. The process of finding the integral is called integration. Each of these indefinite integrals may be verified by differentiating the right hand side and verifying that the integrand is the result. Integration Review 5 1.4 Integration by Substitution The last two general integral results allow us to break up an integral of sums or differences into integrals of the individual pieces and to pull out any constant multipliers. Another useful way of solving an integral is to use the Substitution Rule which arises by working the differentiation Chain Rule in reverse. Theorem: 1.2. Substitution Rule (Indefinite Integrals): Suppose u = g(x) is a differentiable function whose range of values is an interval I upon which a further function f is continuous, then Z Z f(g(x))g0(x) dx = f(u) du : where the right hand integral is to be evaluated at u = g(x) after integration. Here the du appearing on the right side is the differential: du = g0(x)dx du 0 which, recall, can be remembered by thinking dx = g (x) and multiplying both sides by dx. When using the Substitution Rule with definite integrals we can avoid the final back-substitution of u = g(x) of the indefinite case by instead just changing the limits of the integral appropriately to the u-values corresponding to the x-limits: Theorem: 1.3. Substitution Rule (Definite Integrals): Suppose u = g(x) is a differentiable function whose derivative g0 is continuous on [a; b] and a further function f is continuous on the range of u = g(x) (evaluated on [a; b]), then Z b Z g(b) f(g(x))g0(x) dx = f(u) du : a g(a) 6 1.5 Integration Examples 1.5 Integration Examples Examples: Evaluate the following integrals: Z p 2 1. t2 + t − dt t2 1 Z 2 2. t2 + 1 dt 0 Z 1 3. x2 x3 + 2 3 dx π Z 4 4. (sec x − tan x) sec x dx 0 p Z cos x 5.