Advanced Mathematics D Calculus Advanced Mathematics D

Chapter Six Intergration

The Area Problem Given a function f that is continuous and nonnegative on an interval [a,b], find the area between the graph of f and the interval [a,b] on the x-axis

Antiderivative Definition A function F is called an antiderivative of a function f on a given interval I if F’(x) =f(x) for all x in the interval.

Antiderivatiation & Integration If Then Or

Indefinite Integral Express is called indefinite integral. It emphasizes that the result of antidifferentiation is “generic” function with a indefinite constant term “ “ is called an integral sign “f(x)” is called integrand “C” is called the constant of integration. dx is the differential symbol, serves to identify the independent variable

Properties of the Indefinite Integral Theorem Suppose that F(x), G(x) are antiderivatives of f(x),g(x), respectively, and c is a constant. Then A constant factor can be moved through an integral sign: An antiderivation of a sum (difference) is the sum (difference) of the antiderivatioves:

Integration Formula

Integration by Substitution Step 1 Look for some composition f(g(x)) within the integrand for which the substitution Produces an integral that is expressed entirely in terms of u and du, (may not OK) Step 2 If OK in Step 1, then try to evaluate the resolution integral in terms of u, (may not OK) Step 3 If OK in Step 2, then replace u by g(x) to express your final answer in term of x.

Difinite Integral – Riemann Sum A function f is said to be integrable on a finite closed interval [a,b] if the limit exists and does not depend on the choice of partitions or on the choice of the points in the subintervals. When this is the case we denote the limit by the symbol which is called the definite integral of f from a to b. The numbers a and are called the lower and upper limit of integration respectively and f(x) is called the integrand.

Difinite Integral – Graph Area Theorem If a function f is continuous on an interval [a,b], then f is integrable on [a,b], and net signed area A between the graph of f and the interval [a,b] is

Definite Integral - Basic Properties Definition If a is in the domain of f, we define If f is integrable on [a,b], then we define

Theorem If f and g are integrable on [a,b] and if c is a constant, then cf, f±g are integrable on [a,b] and

Theorem If f and g are integrable on a closed interval containing the three points a,b and c, then

Theorem If f and g are integrable on [a,b] and f(x)≥0 for all x in [a,b], then If f and g are intgraable on [a,b] and f(x) ≥ g(x) for all x in [a,b], then

Definition A function f that is defined on an interval I is said to be bounded on I if there is a positive number M such that for all x in the interval I. Geometrically, this means that the graph of f over the interval I lied between the lines y=-M and y=M

Theorem Let f be a function that is defined on the finite closed interval [a,b] If f(x) has finitely many discontinuities in [a,b] but bounded on [a,b], then f is integrable on [a,b] If f is not bounded on [a,b], then f(x) is not integrable on [a,b]

The Fundamental Theorem of Calculus Part 1 If f is continuous on [a,b] and F is any antiderivative of f on [a,b], then

Relationship between Indefinite and definite Integration

Dummy Variable Variable of integration in a definite integral plays no role in the end result So you can change the variable of the integration whenever you feel it convenient

The Mean-Value Theorem for Integral If f is continuous on a closed interval [a,b], then there is at least one point x* in [a,b] such that

The Fundamental Theorem of Calculus Part 2 If f is continuous on an interval I, then f has an antiderivative on I. In particular if a is any point in I, then the function F defined by is an antiderivative of f on I; that is

Integration A Rate of Change Integrating the rate of change of F(x) with respect to x over an interval [a,b] products the change in the value of F(x) that occurs as x increases from a to b.

Evaluation Definite Integrals by Substitution Method1 Calculate the indefinite integral by substitution Using fundemantal Th. of Calculus Method2 Direct substitute all variable including the upper and lower limitations

Theorem If g’ is continuous on [a,b] and f is continuous on an interval containing the values of g(x) for a≤x≤b, then

First Area Problem Suppose that f and g are continuous functions on an interval [a,b] and Find the area A of the region bounded above by y= f(x), below by y= g(x), and on the sides by the lines x=a and x=b.

Area Formula If f and g are continuous functions on the interval [a,b], and if f(x) ≥ g(x) for all x in [a,b], then the area of the region bounded above by y = f(x), below by y = g(x), on the left by the line x=a and on the right by the line x=b is

Finding the Limits of Integraton for the Area between Two Curves Step 1 Sketch the region Step 2 The y-coordinate of the top end point of the line segment sketched in Step 1 will be f(x),the bottom one g(x), and the length of the line segment will be integrand f(x) - g(x) Step 3 Determine the limits. The left at which the line segment intersects the region is x=a and the right most is x=b.

Second Area Problem Suppose that w and v are continuous functions of y on an interval [c,d] and that Find the area A of the region bounded on the left by x=v(y), right by x=w(y), and on below by the y=c and y=d.