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1、Chapter 4 Integration4.1 Indefinite Integrals Indefinite Integrals An indefinite integral of f(x) represent an entirely family of functions whose derivative is f(x) and is denoted byThen:means Example 1:BecauseExample 2:BecauseTable of Indefinite integrals ( C is a constant)ororThe properties of Ind
2、efinite IntegralsCorollary: IfthenExample 3: Solution:Example 4:Solution: =Example 5:Solution: =Example: Solution: =4.2 The Substitution RuleThe Substitution RuleExample:LetthenLetthen The Substitution Rule:Suppose( ) The Substitution Rule:If is a differentiabl function whose rangeIs an interval I a
3、nd is continuous on I,then Example 1 Notice that at the final stage we had to returnto the original variable xExample 2 Example 3Example 4 Example 5 (1).(2). (3 ).Ex6Ex7ExampleExample4.3 Estimating with finite sums The area problem Area of rectangleArea of trapezoid 1. The area problem 1. Ares of cu
4、rved trapezoidSuppose the curved trapezoid is bounded byandFind the area of A .Figure 1S2. Distance traveledSuppose v=v(t) is continuous on a,b.How far the object traveled on the time interval a,b ?3. The average value of continuous function4.4 The definition of integralThe definite integral as a li
5、mit of Riemann sumsLet f be a function defined on a closed interval a,b. For any partition P of a,b, let the number ck be chosen arbitrarily in the subinterval xk-1,xk. If there exists a number I such thatno matter how P and the ck are chosen ,the f is integrable on a,b and I is the the definite int
6、egral of f over a,b.where:integral signf(x) : integranda : lower limit of integrationb : upper limit of integration The procedure of calculating an integral is called integration.Theorem1.Theorem2.and have only finite discontinuous on a,b The sufficient condition of integrability:is integral on is i
7、ntegral on Evaluating IntegralsFormulasExample (a) Evaluate the Riemann sum for taking the sample points to be rightendpoints and (b) EvaluateSolution(a) With the interval width is And the right end points areSo the Riemann sum is(a) With subintervals we have Using the right endpoints ,we getExample
8、 1. Use the definition to evaluateSolution: Useto divide 1, 3 into n subintervals of equal width.Caution:The definite integral is a number; it does not depend on x. In fact ,we could use and letter in place of x without changing the value of the integral: where x is a dummy variable.Geometric interp
9、retation:A is the area of the regionThe negative of the area of the region:the algebraic sum of the oriented area, or the net area.Solution:Use the integral to denote the following limits:Solution:Use the integral to denote the following limits:Use the geometric meaning to evaluate:andFind the total
10、 distance s.Method:1) Partition:2) Approximation:We get SupposeSuppose the distance overThe Distance Problem3) Sum:4) Limit:The Midpoint RuleProperties of the definite integralWhen we defined the definite integral . we implicitly assumed that ab.Notice that if we reverse a and b ,then changefrom (b-
11、a)/n to (a-b)/n .ThereforeProperties of the integral( k is a constant)Proof:( c is a constant) Example: Use the properties of integrals to evaluate Comparison Properties of the integral:SolutionUse properties of integral to estimate the valueof integralLetThen4.5 The Fundamental Theorem of CalculusI
12、f is the velocity of an object and is itsPosition at time , thenand the distance (displacement) is The Fundamental Theorem of Calculus The fundamental Theorem of Calculus is appropriately named because it establishes a connection between the two branches to Calculus: differential calculus and integr
13、al calculus. Differential calculus arose from the tangent problem,whereas integral calculus arose from a seemingly unrelated problem ,the area problem.The Fundamental Theorem of Calculus gives the precise relationship between the derivative and the integral.Newtons teacher at Cambridge, Issac Barrow
14、 discovered that the two problem are actually closely related. In fact ,he realized that differentiation and integration are inverse processes.It was Newton and Leibniz who exploited this precise relationship and use it to develop calculus into a systematic mathematical method.In particular, they sa
15、w that the Fundamental Theorem enabled them to compute areas and integrals very easily without having to compute them as limits of sums.The first part of the Fundamental Theorem deals with functions defined by an equation of the formwhere f is a continous function on a,b and x varies between a and b
16、 .Observe that g depends only on x ,which appears as the variable upper limit in the integral.If x is a fixed number ,then the integral is a definite number.If we then let x vary ,the number also varies and defines a function of x by g(x).baxf(x)The fundamental Theorem of Calulus, Part IIf is contin
17、uous on ,thenthe function defined byis continuous on and diffe-rentiable on andThe fundamental Theorem of Calulus, Part IProof:thenSince f is continuous on x,x+h,the Extreme ValueTheorem says that there are number u and v inx,x+h such that f(u)=m and f(v)=M,where m andM are the absolute minimun and
18、maximum valuesof f on x,x+hWe have By the squeeze theorem,we haveExampleThe Fundamental Theorem of Calculus, Part 2Proof:According part 1,sosowhere F is any antiderivativev of f ,that is,a function such that F=fwe havedenoteWhy?Summary:We end this section by bringing together the two parts of the Fu
19、ndamental Theorem.Definite IntegralsThe Substitution Rule for Definite IntegralsIf is continuous on a, b and f is continuous on the range of u=g(x), thenProof: Suppose F be an antiderivative of f. Then F(g(x) is an antiderivative of SolutionSolutionSolutionSolutionEx3.Proof:(1) If(2) IfOdd functionSolution- -+ +12112 - -12dxxxEven function ExampleDefinition The natural logarithmic function is the function defined byBased on the above definition of natural logarithmic function , we will deduce all properties of logarithm functions。As far as the details ,we left to you as an exercise.
