Presentation on theme: "Chapter 6 The Integral Sections 6.1, 6.2, and 6.3"— Presentation transcript:
1 Chapter 6 The Integral Sections 6.1, 6.2, and 6.3
2 The Integral The Indefinite Integral Substitution The Definite Integral As a SumThe Definite Integral As Area
3 IntroductionA physicist who knows the velocity of a particle might wish to know its position at a given time.A biologist who knows the rate at which a bacteria population is increasing might want to deduce what the size of the population will be at some future time.
4 Antiderivatives Definition In each case, the problem is to find a function F whose derivative is a known function f.If such a function F exists, it is called an antiderivative of f.DefinitionA function F is called an antiderivative of f onan interval I if F’(x) = f (x) for all x in I.
5 Antiderivatives For instance, let f (x) = x2. It is not difficult to discover an antiderivative of f if we keep the Power Rule in mind.In fact, if F(x) = ⅓ x3, then F’(x) = x2 = f (x).
6 AntiderivativesHowever, the function G(x) = ⅓ x also satisfies G’(x) = x2.Therefore, both F and G are antiderivatives of f.Indeed, any function of the form H(x)=⅓ x3 + C, where C is a constant, is an antiderivative of f.The question arises: Are there any others?
7 AntiderivativesTheoremIf F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is F(x) + Cwhere C is an arbitrary constant.
8 AntiderivativesGoing back to the function f (x) = x2, we see that the general antiderivative of f is ⅓ x3 + C.
9 Family of FunctionsBy assigning specific values to C, we obtain a family of functions.Their graphs are verticaltranslates of one another.This makes sense, as eachcurve must have the sameslope at any given valueof x.
10 Notation for Antiderivatives The symbol is traditionally used to represent the most general an antiderivative of f on an open interval and is called the indefinite integral of f .Thus, means F’(x) = f (x)
11 Indefinite Integral The expression: read “the indefinite integral of f with respect to x,” means to find the set of all antiderivatives of f.x is called the variable of integrationIntegrandIntegral sign
12 Indefinite Integral For example, we can write Thus, we can regard an indefinite integral as representing an entire family of functions (one antiderivative for each value of the constant C).
13 Constant of Integration Every antiderivative F of f must be of the form F(x) = G(x) + C, where C is a constant.Example:Represents every possible antiderivative of 6x.
14 Power Rule for the Indefinite Integral Example:
15 Power Rule for the Indefinite Integral Indefinite Integral of ex and bx
25 The Definite IntegralLet f be a continuous function on [a, b]. If F is any antiderivative of f defined on [a, b], then the definite integral of f from a to b is defined byis read “the integral, from a to b of f (x) dx.”
26 Notation In the notation , f (x) is called the integrand. a and b are called the limits of integration; a is the lower limit and b is the upper limit.For now, the symbol dx has no meaning by itself; is all one symbol. The dx simply indicates that the independent variable is x.
27 The Definite IntegralThe procedure of calculating an integral is called integration. The definite integral is a number. It does not depend on x.Also note that the variable x is a “dummy variable.”
28 Geometric Interpretation of the Definite Integral The Definite Integral As AreaThe Definite Integral As Net Change of Area
29 Definite Integral As Area If f is a positive function defined for a ≤ x ≤ b, then the definite integral represents the area under the curve y = f (x) from a to b
30 Definite Integral As Area If f is a negative function for a ≤ x ≤ b, then the area between the curve y = f (x) and the x-axis from a to b, is the negative of
31 Definite Integral As Area Consider y = f (x) = 0.5x + 6 on the interval [2,6] whose graph is given below,
32 Definite Integral As Area Consider y = f (x) = 0.5x + 6 on the interval [2,6] whose graph is given below,
33 Definite Integral as Net Area If f changes sign on the interval a ≤ x ≤ b, then definite integral represents the net area, that is, a difference of areas as indicated below:a bR1R2R3Area of R1 – Area of R2 + Area of R3
34 Total AreaIf f changes sign on the interval a ≤ x ≤ b, and we need to find the total area between the graph and the x-axis from a to b, thenTotal Area Area of R1 + Area of R2 + Area of R3Area of R1a bR1R2R3Area of R2cdArea of R3
35 Area Using Geometry Example: Use geometry to compute the integral –15Area = 8Area = 2
36 Area Using Antiderivatives Example: Use an antiderivative to compute the integralFirst, we need an antiderivative of
37 Area Using Antiderivatives Example: Now find the total area bounded by the curve and the x-axis from x –1 to x 5.R2–115R1Total Area Area of R1 + Area of R2
39 Evaluating the Definite Integral Example: Calculate
40 Substitution for Definite Integrals Example: CalculateNotice limits change
41 Computing AreaExample: Find the area enclosed by the x-axis, the vertical lines x = 0, x = 2 and the graph ofGives the area since 2x3 is nonnegative on [0, 2].Antiderivative
42 The Definite Integral As a Total If r (x) is the rate of change of a quantity Q (in units of Q per unit of x), then the total or accumulated change of the quantity as x changes from a to b is given by
43 The Definite Integral As a Total Example: If at time t minutes you are traveling at a rate of v(t) feet per minute, then the total distance traveled in feet from minute 2 to minute 10 is given by