Chapter 5 Integration Section R Review

Chapter 5 Review Important Terms, Symbols, Concepts 5.1 Antiderivatives and Indefinite Integrals A function F is an antiderivative of a function f if F ´(x) = f (x) If F and G are both antiderivatives of f, they differ by a constant: F(x) = G(x) + k for some constant k. We use the symbol , called an indefinite integral, to represent the family of all antiderivatives of f, and we write

Chapter 5 Review 5.1 Antiderivatives and Indefinite Integrals (continued) The symbol is called an integral sign, f (x) is the integrand, and C is the constant of integration. Indefinite integrals of basic functions are given in this section. Properties of indefinite integrals are given in the section; in particular, a constant factor can be moved across an integral sign. However, a variable factor cannot be moved across an integral sign.

Chapter 5 Review 5.2 Integration by Substitution The method of substitution (also called the change-of-variable method) is a technique for finding indefinite integrals. It is based on the following formula, which is obtained by reversing the chain rule: This formula implies the general indefinite integral formulas in this section. Guidelines for using the substitution method are given by the procedure in this section.

Chapter 5 Review 5.2 Integration by Substitution (continued) In using the method of substitution it is helpful to employ differentials as a bookkeeping device: The differential dx of the independent variable x is an arbitrary real number. The differential dy of the dependent variable y is defined by dy = f ´(x) dx. 5.3 Differential Equations; Growth and Decay An equation is a differential equation if it involves an unknown function and one or more of its derivatives.

Chapter 5 Review 5.3 Differential Equations; Growth and Decay (continued) The equation is a first-order differential equation because it involves the first derivative of the unknown function y, but no second or higher-order derivative. A slope field can be constructed for the differential equation above by drawing a tangent line with slope 3x(1 + xy2) at each point (x, y) of a grid. The slope field gives a graphical representation of the functions that are solutions of the differential equation.

Chapter 5 Review 5.3 Differential Equations; Growth and Decay (continued) The differential equation (in words: the rate at which the unknown function Q increases is proportional to Q) is called the exponential growth law. The constant r is called the relative growth rate. The solutions to the exponential growth law are the functions Q(t) = Q0ert where Q0 denotes Q(0), the amount present at time t = 0. These functions can be used to solve problems in population growth, continuous compound interest, radioactive decay, blood pressure, and light absorption.

Chapter 5 Review 5.3 Differential Equations; Growth and Decay (continued) A table in this section gives the solutions to other first-order differential equations used to model the limited or logistic growth of epidemics, sales and corporations. 5.4 The Definite Integral If the function f is positive on [a, b], then the area between the graph of f and the x axis from x = a to x = b can be approximated by partitioning [a, b] into n subintervals [xk–1, xk ] of equal length ∆x = (b – a)/n, and summing the areas of n rectangles.

Chapter 5 Review 5.4 The Definite Integral (continued) The process of summing the areas of n rectangles can be accomplished by left sums, right sums, or, more generally, by Riemann sums: Left sum Right sum Riemann sum In a Riemann sum, each ck is required to belong to the subinterval [xk-1, xk ]. Left sums and right sums are special cases of Riemann sums.

Chapter 5 Review 5.4 The Definite Integral (continued) The error in an approximation is the absolute value of the difference between the approximation and the actual value. An error bound is a positive number such that the error is guaranteed to be less than or equal to that number. Theorem 1 in this section gives error bounds for the approximation of the area between the graph of a positive function f and the x axis, from x = a to x = b, by left sums or right sums, if f is either increasing or decreasing.

Chapter 5 Review 5.4 The Definite Integral (continued) If f (x) > 0 and is either increasing or decreasing on [a, b], then its left and right sums approach the same real number I as n → ∞. If f is a continuous function on [a, b], then the Riemann sums for f on [a, b] approach a real number limit I as n → ∞. Let f be a continuous function on [a, b]. The limit I of Riemann sums for f on [a, b] is called the definite integral of f from a to b, denoted

Chapter 5 Review 5.4 The Definite Integral (continued) The integrand is f (x), the lower limit of integration is a, and the upper limit of integration is b. Geometrically, the definite integral represents the cumulative sum of the signed areas between the graph of f and the x axis from x = a to x = b. Properties of the definite integral are given in this section.

Chapter 5 Review 5.5 The Fundamental Theorem of Calculus If f is a continuous function on [a, b] and F is any antiderivative of f, then The fundamental theorem gives an easy and exact method for evaluating definite integrals, provided we can find an antiderivative F(x) of f (x). In practice, we first find an antiderivative F(x) (when possible) using techniques for computing indefinite integrals, then we calculate F(b) – F(a). If it is impossible to find an antiderivative we must resort to left or right sums or other approximation methods.

Chapter 5 Review 5.5 The Fundamental Theorem of Calculus Graphing calculators have built-in numerical approximation routines, more powerful than left or right sum methods, for calculating the definite integral. If f is a continuous function on [a, b], then the average value of f over [a, b] is defined to be