微積分第五章課件 Integrals

1、Chapter 5 Integrals5.1 Areas and Distance5.2 The Definite Integral5.3 The Fundamental Theorem of Calculus5.4 Indefinite Integrals and the Net Change Theorem5.5 The Substitution Rule5.6 The Logarithm Defined as an Integral5.1 Areas and Distances The area problem 1. Areas of curved trapezoidSuppose th

2、e curved trapezoid is bounded byandFind the area of A .Area of rectangleArea of trapezoidMethod:1) Partition:Use the linesto divide A into small curved trapezoid;2) Approximation:Base:Height:We can approximateSmall rectangle:3) Sum:4) Limit:Letthen the area of the curved trapezoid is:Riemann sum2. D

3、istanceandFind the total distance s.Method:1) Partition:2) Approximation:We get SupposeSuppose the distance over3) Sum:4) Limit:Common characteristic of the above problems is:1) The process is the same :“Partition, Approximation, Sum, Limit ”2) The limit forms are the same: Both are the limits of Ri

4、emann sumsRiemann sum5.2 The definition of integral The definition of integralalways tends to I , We say that I is the integral of That is Then we say that f ( x ) is integrable on a , b .denote it bywhere:integral signf(x) : integranda : lower limit of integrationb : upper limit of integration The

5、procedure of calculating an integral is called integration.Caution:The definite integral is a number; it does not depend on x. In fact ,we could use any letter in place of x without changing the value of the integral: where x is a dummy variable.Geometric interpretation:the area of the region:the ne

6、gative of the area of the region:the algebraic sum of the oriented area, or the net area.Theorem1.Theorem2.and have only finite discontinuities on a,b The sufficient condition of integrability:is integrable on is integrable on Example 1. Use the definition to evaluateSolution:UseWe choose to divide

7、0, 1 into n subintervals of equal width.Example 2. Use the definition to evaluateSolution:UseWe choose to divide 1, 3 into n subintervals of equal width.Solution:Use the integral to denote the following limits:Use the geometric meaning to evaluate:The Midpoint RuleProperties of the definite integral

8、When we defined the definite integral . we implicitly assumed that ab.Notice that if we reverse a and b ,then change (b-a)/n from 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 th

9、e integral:The next theorem is called Mean Value Theorem for Definite Integrals. Its geometric interpretation is that, for a continuous positive f(x) on a, b, there is a number c in a, b such that the rectangle with base a, b and height f(c) has the same area as the region under the graph of f(x) fr

10、om a to b.oabxycf(x)Theorem If f(x) is continuous on a, b, then there exists at least a number c in (a, b) such thatThe number is called the average value of f(x) on a, b.Proof If f(x) is a constant function, the result is true. Next we assume that f(x) is not a constant function. Since f(x) is cont

11、inuous on a, b, f(x) takes on the minimum and the maximum values on a, b. Let f(u) = m and f(v) = M be the minimum and the maximum values of f(x) on a, b, respectively. Then m f(x) M for some x in a, b because f(x) is not constant. Therefore, we have It follows that The preceding inequalities indica

12、te that the number is between m =f(u) and M =f(v). Thus the Intermediate Value Theorem leads that there is a number c between u and v such that Multiplying the both sides by b-a gives the conclusion of the theorem. 5.3 The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus The funda

13、mental Theorem of Calculus is appropriately named because it establishes a connection between the two branches to Calculus: differential calculus and integral calculus. Differential calculus arose from the tangent problem, whereas integral calculus arose from a seemingly unrelated problem ,the area

14、problem.The Fundamental Theorem of Calculus gives the precise relationship between the derivative and the integral.Newtons teacher at Cambridge, Issac Barrow discovered that the two problem are actually closely related. In fact ,he realized that differentiation and integration are inverse processes.

15、It was Newton and Leibniz who exploited this precise relationship and use it to develop calculus into a systematic mathematical method.In particular, they saw that the Fundamental Theorem enabled them to compute areas and integrals very easily without having to compute them as limits of sums.The fir

16、st part of the Fundamental Theorem deals with functions defined by an equation of the formwhere f is a continuous function on a,b and x varies between a and b .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 d

17、efinite number.If we then let x vary ,the number also varies and defines a function of x by g(x).baxf(t)The Fundamental Theorem of Calculus, Part IProof:thenThe Fundamental Theorem of Calculus, Part 2Proof:According to part 1,sosowe havedenoteWhy?Summary:We end this section by bringing together the

18、two parts of the Fundamental Theorem.5.4 Indefinite Integrals and the Net Change Theorem Indefinite Integrals and the Net Change TheoremAn indefinite integral of f(x) represent an entirely family of functions whose derivative is f(x) and is denoted bySuppose that F(x) is an antiderivative of f(x), t

19、hen according to the theorem in 4.2, we know that any antiderivative G(x) of f(x) can be written as G(x)=F(x)+CThen:Caution: 1) You should distinguish carefully between definite and indefinite integrals. A definite integral is a number, whereas an indefinite integral is a function (a family of funct

20、ions). 2) The connection is the fundamental theorem:Table of Indefinite integrals From the above definition ,we known that:oror( k is a constant)ororExample 1:Sulotion: = Example 2: Solution: =The properties of Indefinite IntegralsCorollary: IfthenExample 3: Solution: =Example 4:Solution: =Example 5

21、:Solution: =Example: Solution: =Application:The Net Change Theorem The integral of a rate of change is the net change:Example 6 A particle moves along a line so that its velocity at time t is 1) Find the displacement of the particle during the time period 2)Find the distance traveled during the time

22、 period.Solution:5.5 The Substitution RuleThe Substitution RuleExample: The Substitution Rule:Suppose( )Exampe 1 Example 2 Example 3Example 4 Example 5 (1).(2). (3 ).Ex6Ex7Definite 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 antid