bcb12e ppt 6 R

Chapter 6

Integration

Chapter 6 Review

Important Terms, Symbols, Concepts

 _{6.1 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

( )

f x dx

∫

Chapter 6 Review

 _{6.1 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 6 Review

 _{6.2 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.

′

f

[

g

(

x

)]

g

′

(

x

) =

f

[

g

(

x

)] +

C

Chapter 6 Review

 6.2 Integration by Substitution (continued)

• In using the method of substitution it is helpful to employ differentials as a bookkeeping device:

o The differential dx of the independent variable x is an arbitrary real number.

o The differential dy of the dependent variable y is defined by dy = f ’(x) dx.

 6.3 Differential Equations; Growth and Decay

Chapter 6 Review

 _{6.3 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.

2 3 (1 )

dy

x xy

Chapter 6 Review

 _{6.3 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 Q

0 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

dQ

Chapter 6 Review

 _{6.3 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.

 _{6.4 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

Chapter 6 Review

 _{6.4 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 c_k is required to belong to the subinterval

[x_k-1, x_k]. Left sums and right sums are special cases of Riemann sums.

Chapter 6 Review

 6.4 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

Chapter 6 Review

 _{6.4 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

Chapter 6 Review

 _{6.4 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.

( )

f x dx

Chapter 6 Review

 _{6.5 The Fundamental Theorem of Calculus}

• If f is a continuous function on [a, b] and F is any antiderivative of f, then = F(b) – F(a)

• 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

( )

f x dx

Chapter 6 Review

 _{6.5 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

1

( )

b

a