第6.7节 反常积分

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6.7IMPROPER INTEGRALS

Whatis the area of the region under the curve y=1/______fromx=0 to x=1(Figure 6.7.1(a)) ? The function 1/_____isnot continuous at x=0, and in fact 1/_____isinfinite for infinitesimal ε>0.Thus our notion of a definite integral does not apply. Neverthelesswe shall be able to assign an area to the region using improperintegrals. We see from the figure that the region extends infinitelyfar up in the vertical direction. However, it becomes so thin thatthe area of the region turns out to be finite.

Theregion of Figure 6.7.1(b)under the curve y=x -3fromx=1to x=∞

Figure6.7.1

extendsinfinitely far in the horizontal direction. We shall see that thisregion, too, has a finite area which is given by an improperintegral.

Improperintegrals are defined as follows.

DEFINITION

Supposefis continuous on the half - open interval [a, b]. The improperintegral of f from a to b is defined by the limit

Ifthe limit exists the improper integral is said to converge. Otherwisethe improper integral is said to diverge.

Theimproper integral can also be described in terms of definiteintegrals with hyperreal endpoints. We first recall that the definiteintegral

isa real function of two variables uandv.If u andv varyover the hyperreal numbers instead of the real numbers, the definiteintegral ____f(x)dx standsfor the natural extension of Devaluatedat (u,v),

Hereis description of the improper integral using definite integrals withhyperreal endpoints.

Letf becontinuous on (a, b].

(1)___f(x)dx=S ifand only if __ f(x) dx ≈ S for all positive infinite ε.

(2)______f(x) dx = ∝ (or -∝ ) if andonly if ___ f(x) d(x) is positive infinite (or negative infinite) forall

positiveinfiniteε.

EXAMPLE1 Find ________.Foru >0,

Then

Thereforetheregion under the curve y=1/____from 0 to 1 shown in Figure 6.7.1(a) has area 2,

andtheimproper integral converges.

EXAMPLE2Find________dx.For u >0,

Thistime

Theimproper integral diverges. Since the limit goes to infinity we maywrite

______x-2dx =∞

Theregion under the curve in Figure 6.7.2 is said to have infinitearea.

Warning:we remind the reader once again that the symbols ∞and-∞arenot real or even hyperreal numbers. We use them only to indicate thebehavior of a limit, or to indicate an interval without an upper orlower endpoint.

6.7.2

EXAMPLE3 Find the length of the curve y=x2/3,0 ≤ x ≤8.From Figure 6.7.3 the curve must have finite length. However, thederivative

____________________

isundefined at x=0.Thus the length formula gives an improper integral,

Figure6.7.3

Letu=9x2/3+4, du = 6x-1/3dx.The indefinite integral is

Therefore

Noticethat we use the same symbol for both the definite and the improperintegral. The theorem below justifies this practice.

THEOREM1

Iff is continuous on the closed interval [a,b] then the improperintegral of f from a to b converges an equals the definite integralof f from a to b.

PROOFWe have shown in Section 4.2 on the Fundamental Theorem that thefunction

F(u)= ____ f(x)dx

iscontinuous on [a, b].Therefore

where___f(x)dxdenotesthe definite integral.

Wenow define a second kind of improper integral where the interval isinfinite.

DEFINITION

Letf be continuous on the half-open interval[a,∝). The improperintegral of f from a to ∝ is defined by the limit

Theimproperintegral is said to converge if the limit exists and to divergeotherwise.

Hereis description of this kind of improper integral using definiteintegrals with hyperreal endpoints.

Letfbe continuous on [a,∝).

(1)___f(x)dx =S ifand only if ___f(x) dx ≈ Sfor all positive infinite H.

(2)___f(x)dx =∞(or- ∞)if and only if___f(x) dx is positive infinite (ornegative infinite ) for all positive

infiniteH.

EXAMPLE4 Findthe area under the curve y= x-3from1 to ∝.The area is given by the improper integral

For

Thus

Sothe improper integral converges and the region has area___.The region is shown in Figure 6.7.1(b)and extends infinitely far to the right.

EXAMPLE5 Findthe area under the curve y=x-2/3,1≤ x≤∝.

Theregion is shown in Figure 6.7.4 and has infinite area.

Figure6.7.4

EXAMPLE6

Theregionin Example 5 is rotated about the x-axis.Find the volume of the solid of revolution.

Weusethe disc method because the rotation is about the axis of theindependent variable. The volume

formulagivesus an improper integral.

Sothesolid shown in Figure 6.7.5 has finite volume V=3π.

Figure6.7.5

Thelast two examples give an unexpected result. A region with infinitearea is rotated about the x-axisand generates a solid with finite volume! In terms of hyperrealnumbers, the area of the region under the curve y=x-2/3from1 to an infinite hyperreal number Hisequal to 3( H1/3-1),which is positive infinite. But the volume of the solid of revolutionfrom 1 to Hisequal to

3π(1- H-1/3),

Whichis finite and has standard part 3π.

Wecan give a simpler example of this phenomenon. Let Hbea positive infinite hyperinteger, and form a cylinder of radius 1/Handlength H²(Figure6.7.6). Then the cylinder is formed by rotating a rectangle of lengthH², width 1/H,and infinite area H²/H=H.But the volume of the cylinder is equal to π,

V=π r²h = π(1/H)²(H)²=π.

Imaginea cylinder made out of modelling clay, with initial length and radiusone. The volume is π. The clay is carefully stretched so that thecylinder gets longer and thinner. The volume stays the same, but thearea of the cross section keeps getting bigger. When the lengthbecomes infinite, the cylinder of clay still has finite volume V=π,but the area of the cross section has become infinite.

Thereare other types of improper integrals. If f is continuous onthe half - open interval [a,b] then we define

_______f(x)dx =________f(x)dx.

Iff is continuous on (-∝,b] we define

_______f(x)dx =________f(x)dx.

Wehave introduced four types of improper integrals corresponding to thefour types of half - open intervals

[a,b),[a,∞), (a,b], ( - ∞,b].

Bypiecing together improper integrals of these four types we can assignan improper integral to most functions which arise in calculus.

DEFINITION

Afunction f is said to be piecewise continuous onan interval I if f is defined and continuous at all but perhapsfinitely many points of I. In particular, every continuous functionis piecewise continuous.

Wecan introduce the improper integral ____f(x)dx whenever f is piecewise continuous on I anda,b are either the endpoints of I or the appropriateinfinity symbol. A few examples will show how this can be done.

Letf be continuous at every point of the closed interval[a,b]except at one point c where a<c<b. We define

_______f(x)dx= _____f(x)dx+ ____ f(x)dx.

EXAMPLE7

Findtheimproper integral ____ x -1/3 dx.x -1/3 is discontinuous at x=0. The indefinite integral is

∫ x-1/3dx = ___ x2/3+ C.

Then

Similarly,

So

andthe improper integral converges. Thus, the region shown inFigure6.7.7 has finite area.

Figure6.7.7

Iff is continuous on the open interval (a, b), theimproper integral is defined as the sum

wherec is any point in the interval (a,b). The endpoints aand b may be finite or infinite. It does not matter whichpoint c is chosen, because if e is any other point in(a,b), then

EXAMPLE8 Find

Thefunction 2/______ +1/____is continuous on the open interval (0, 2)but discontinuous at bothendpoints (Figure 6.7.8). Thus

Figure6.7.8

Firstwe find the indefinite integral.

Then

Also

Therefore

EXAMPLE9 Find

Thefunction 1/x² + 1/(x-1)² is continuous on the openinterval (0,1) but discontinuous at both endpoints. The indefiniteintegral is

Wehave

Similarlywe find that

Inthis situation we may write

andwe say that the region under the curve in Figure 6.7.9 has infinitearea.

Figure6.7.9

RemarkIn Example 9

Weare faced with a sum of two infinite limits. Using the rules foradding infinite hyperreal

numbersas a guide we can give rules for sums of infinite limits.

IfH and K are positive infinite hyperreal numbers and cis finite, then

H+ K is positive infinite,

H+ c is positive infinite,

-H- K is negative infinite,

-H+ c is negative infinite,

H-Kcan be either finite, positive infinite, or negative infinite.

Byanalogy, we use the following rules for sums of two infinite limitsor of a finite and an infinite limit. These rules tell us when such asum can be considered to be positive or negative infinite. We use theinfinity symbols as a convenient shorthand, keeping in mind that theyare not even hyperreal numbers.

EXAMPLE10 Find _____ xdx. We see that

and______xdx = ∞.

Thus____xdx diverges and has the form ∞-∞. We do not assign it anyvalue or either of the symbols ∞ or -∞. The region under thecurve f(x)=x is shown in Figure 6.7.10.

Figure6.7.10

Itis tempting to argue that the positive area to the right of theorigin and the negative area to the left exactly cancel each otherout so that the improper integral is zero. But this leads to aparadox.

Wrong:_____ xdx =0. Let v= x+2,dv= dx. Then

Subtracting

But______ 2dx=∞

Sowe do not give the integral_____ xdx the value 0, and insteadleave it undefined.

PROBLEMSFOR SECTION 6.7

InProblems 1-36, test the improper integral for convergence andevaluate when possible.

1_________x-2dx2________x-0.9dx

3________x -1/2dx 4______(2x-1)-3dx
5______(2x-1)-3dx6_______x -1/3dx

7______x2+2x-1dx8_________x-2- x -3dx

9__________x(1+x2)-2dx 10_______x -1/2+x-2dx

11________x-1/2+x-2dx ??12_________x-2dx

13_______(x-1)-2/3dx14________x-2dx

15_____x-2/3dx16______dx

17____2x(x2-1)-1/3 dx18____2x3dx

19_____(2x-1)-2/3dx20_____(3x-1)-5dx

21______x²dx22______(2x-1)3 dx

23_________dx24_______x-1/3dx

25________ x3dx26_______x-3/2dx

27________dx28______ |x| (x+ 1)-3dx

29_________dx30_______(x-1)-2 + (x-3)-2dx

31____(x-1)-1/2+ (3-x) -1/2 dx32______dx

33__________34____________

35_________f(x)dxwhere f(x)= ______

36_________f(x)dx where f(x)= ______

37Show that if r is a rational number, the improper integral____ x-r dx converges whenr <1

anddivergeswhen r >1.

38Show that if r is a rational, the improper integral ____x-r dx converges when r >1

anddivergeswhen r <1.

39Find the area of the region under the curve y=4x-2from x=1 to x=∝.

40Find the area of the region under the curve y=1/_______from x=_____ to x=1.

41Find the area of the region between the curves y=x-1/4and y=x-1/2from x=0 to x=1.

42Find the area of the region between the curves y=-x-3and y=x -2,1 ≤ x<∝.

43Find the volume of the solid generated by rotating the curve y=1/x,1 ≤ x<∝,

about(a)the x-axis,(b)the y-axis.

44Find the volume of the solid generated by rotating the curve y=x-1/3,0 ≤ x<1,

about(a)the x-axis, (b) the y-axis.

45Find the volume of the solid generated by rotating the curve y=x-3/2,0 ≤ x<4,

about(a)the x-axis, (b) the y-axis.

46Find the volume generated by rotating the curve y=4x-3,-∝ ≤ x< -2, about (a) the x-axis,

(b)they-axis.

47Find the length of the curve y=_____from x=0 to x=1.

48Find the length of the curve y=_____from x=0 to x=1.

49Find the surface area generated when the curve y=___________,0 ≤x ≤1, is rotated about

(a)thex-axis, (b) the y-axis.

50Do the same for the curve y=______, 0 ≤ x ≤1

51(a) Find the surface area generated by rotating the curve y=____ , 0 ≤x ≤1, about thex-axis.

(b)Setupan integral for the area generated about the y-axis.

52Find the surface area generated by rotating the curve y=x2/3,0 ≤x ≤ 8, about the x-axis.

53Find the surface area generated by rotating the curve y=_______,0 ≤x ≤ a, about (a) the

x-axis,(b) the y-axis (0< a ≤ r).

54The force of gravity between particles of mass m1 and m2is F= gm1 m2 /s² where s is the

distancebetweenthem. If m1 is held fixed at the origin, find the work done inmoving m2

fromthepoint (1,0) all the way out the x-axis.

55Show that the Rectangle and Addition Properties hold for improperintegrals.

EXTRAPROBLEMS FOR CHAPTER 6

1The skin is peeled off a spherical apple in four pieces in such a waythat each horizontal cross section is a

squarewhosecorners are on the original surface of the apple. If the originalapple had radius r, find the

volumeofthe peeled apple.

2Find the volume of a tetrahedron of height h and base a righttriangle with legs of length a and b.

3Find the volume of the wedge formed by cutting a right circularcylinder of radius r with two planes, meeting

onaline crossing the axis, one plane perpendicular to the axis and theother at a 45°angle.

4Find the volume of a solid whose base is the region between thex-axis and the curve y=1-x²,and which

intersectseachplane perpendicular to the x-axis in a square.

InProblems 5-8, the region bounded by the given curves is rotated about(a) the x-axis, (b) the y-axis. Find thevolumes of the two solids of revolution.

5y=0,y=________, 0≤ x ≤ 1

6y=0,y=x3/2,0≤ x ≤ 1

7y= x, y= 4-x,0≤ x ≤ 2

8y=xp, y=xq, 0≤ x ≤ 1,where 0< q < p

9The region under the curve y=______,0 ≤ x ≤ 1,where 0< p, isrotated about the x-axis.Find the volume of

thesolidof revolution.

10The region under the curve y=(x²+4)1/3, 0 ≤ x ≤ 2,is rotated about the y-axis.Find the volume of the solid of

Revolution.

11Find the length of the curve y=(2x +1)3/2,0 ≤ x ≤2.

12Find the length of the curve y=3x -2,0 ≤ x ≤4.

13Find the length of the curve x=3t+1,y=2-4t,0 ≤ t ≤1.

14Find the length of the curve x=f(t),y=f(t)+c,a ≤ t ≤b.

15Findthe length of the line x=At+B,y=Ct + D,a ≤ t ≤b.

16Find the area of the surface generated by rotating the curve y=3x²-2,0 ≤ x ≤1,about the y-axis.

17Find the area of the surface generated by rotating the curvex=At²+Bt,y=2At +B,0≤ t ≤ 1,about the x-axis.

A>0,B>0.

18Find the average value of f(x)=x/_______ , 0≤ x ≤4.

19Findthe average value of f(x)=xp,1≤ x ≤b.p≠ -1.

20Find the average distance from the origin of a point on the parabolay=x²,0≤ x ≤ 4.

Withrespect to x.

21Given that f(x) = xp,0≤ x ≤ 1,p apositive constant, find a point cbetween0 and 1 such that f(c)equalsthe average value of f(x)

22Find the center of mass of a wire on the x-axis,0≤ x ≤ 2,whose density at a point xisequal to the square of the distance from (x,0)to (0,1).

23Find the center of mass of a length of wire with constant densitybent into three line segments covering the top, left, and right edgesof the square with vertices (0,0), (0,1), (1,1), (1,0).

24Find the center of mass of a plane object bounded by the linesy=0,y=x,x=1,with density p(x)=1/x.

25Findthe center of mass of a plane object bounded by the curves x=y²,x=1,with density p(x)=y².

26Find the centroid of the triangle bounded by the x-andy-axesand the line ax+by = c,where a, b,and c are

positiveconstants.

27A spring exerts a force of 10x1bswhen stretched a distance x beyondits natural length of 2ft. Find the work required to stretch thespring from a length of 3 ft to 4ft.

InProblems 28-36, test the improper integral for convergence andevaluate if it converges.

28_________x-3dx29____(x+2)-1/4 dx

30____x -4dx31____x -1/5dx

32 ____x 1/5dx33__________dx

34________dx35________dx

36 _____sinx dx

37 Awire has the shape of a curve y=f(x),a ≤x ≤b,and has density p(x)atvalue x.

Justifytheformulas below for the mass and moments of the wire.

38Find the mass, moments, and center of mass of a wire bent in theshape of a parabola

y=x²,-1 ≤ x≤1,with density p(x)=________.

39Find the mass, moments, and center of mass of a wire of constantdensity pbentin the

shapeof the semicircle y=________,-1 ≤x≤1.

□40An object fills the solid generated by rotating the region under thecurve y=f(x),a ≤x ≤b,about the x-axis.

Itsdensity per unit volume is p(x).Justify the following formula for the mass of the object.

m=_____ p(x) π (f(x))²dx.

□41A container filled with water has the shape of a solid of revolutionformed by rotating the curve x=g(y),

a≤y ≤b,about the (vertical) y-axis.Waterhas constant density pperunit volume.

Justifytheformula below for the amount of work needed to pump all the water tothe top of the container.

W=______pπ (g(y))² (b-y) dy.

42Findthe work needed to pump all the water to the top of a water-filledcontainer in the shape of a cylinder

withheighth andcircular base of radius r.

43Do Problem 46 if the container is in the shape of a hemisphericalbowl of radius r.

44Do Problem 46 if the container is in the shape of a cone with itsvertex at the bottom,

heighth,and circular top of radius r.

□45Thepressure,or force per unit area, exerted by water on the walls of a containeris equal to

p=p(b-y)wherep isthe density of water and b-ythewater depth. Find the total force on a dam in the

shapeofa vertical rectangle of height bandwidth w,assumingthe water comes to the top of the dam.

□46Awater-filled container has the shape of a solid formed by rotatingthe curve x=g(y), a ≤y ≤babout

the(vertical)y-axis.Justify the formula below for the total force on the walls of thecontainer.