第6.7节 反常积分

6.7 IMPROPER INTEGRALS

What is the area of the region under the curve y=1/______ from x= 0 to x=1 (Figure 6.7.1(a)) ? The function 1/_____ is not continuous at x= 0, and in fact 1/_____ is infinite for infinitesimal ε >0. Thus our notion of a definite integral does not apply. Nevertheless we shall be able to assign an area to the region using improper integrals. We see from the figure that the region extends infinitely far up in the vertical direction. However, it becomes so thin that the area of the region turns out to be finite.

The region of Figure 6.7.1(b) under the curve y=x -3 from x=1 to x=∞

Figure 6.7.1

extends infinitely far in the horizontal direction. We shall see that this region, too, has a finite area which is given by an improper integral.

Improper integrals are defined as follows.

DEFINITION

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

If the limit exists the improper integral is said to converge. Otherwise the improper integral is said to diverge.

The improper integral can also be described in terms of definite integrals with hyperreal endpoints. We first recall that the definite integral

is a real function of two variables u and v. If u and v vary over the hyperreal numbers instead of the real numbers, the definite integral ____ f(x)dx stands for the natural extension of D evaluated at (u,v),

Here is description of the improper integral using definite integrals with hyperreal endpoints.

Let f be continuous on (a, b].

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

(2) ______ f(x) dx = ∝ (or -∝ ) if and only if ___ f(x) d(x) is positive infinite (or negative infinite) for all

positive infinite ε.

EXAMPLE 1 Find ________. For u > 0,

Then

Therefore the region under the curve y=1/____ from 0 to 1 shown in Figure 6.7.1(a) has area 2,

and the improper integral converges.

EXAMPLE 2 Find ________dx. For u >0,

This time

The improper integral diverges. Since the limit goes to infinity we may write

______ x-2 dx =∞

The region under the curve in Figure 6.7.2 is said to have infinite area.

Warning: we remind the reader once again that the symbols ∞and -∞are not real or even hyperreal numbers. We use them only to indicate the behavior of a limit, or to indicate an interval without an upper or lower endpoint.

6.7.2

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

____________________

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

Figure 6.7.3

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

Therefore

Notice that we use the same symbol for both the definite and the improper integral. The theorem below justifies this practice.

THEOREM 1

If f is continuous on the closed interval [a,b] then the improper integral of f from a to b converges an equals the definite integral of f from a to b.

PROOF We have shown in Section 4.2 on the Fundamental Theorem that the function

F(u) = ____ f(x)dx

is continuous on [a, b]. Therefore

where ___ f(x)dx denotes the definite integral.

We now define a second kind of improper integral where the interval is infinite.

DEFINITION

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

The improper integral is said to converge if the limit exists and to diverge otherwise.

Here is description of this kind of improper integral using definite integrals with hyperreal endpoints.

Let f be continuous on [a,∝).

(1) ___ f(x)dx =S if and only if ___ f(x) dx ≈ S for all positive infinite H.

(2) ___ f(x)dx =∞(or - ∞)if and only if ___ f(x) dx is positive infinite (or negative infinite ) for all positive

infinite H.

EXAMPLE 4 Find the area under the curve y= x-3 from 1 to ∝. The area is given by the improper integral

For

Thus

So the 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.

EXAMPLE 5 Find the area under the curve y= x-2/3, 1≤ x≤ ∝.

The region is shown in Figure 6.7.4 and has infinite area.

Figure 6.7.4

EXAMPLE 6

The region in Example 5 is rotated about the x-axis. Find the volume of the solid of revolution.

We use the disc method because the rotation is about the axis of the independent variable. The volume

formula gives us an improper integral.

So the solid shown in Figure 6.7.5 has finite volume V=3π.

Figure 6.7.5

The last two examples give an unexpected result. A region with infinite area is rotated about the x-axis and generates a solid with finite volume! In terms of hyperreal numbers, the area of the region under the curve y= x-2/3 from 1 to an infinite hyperreal number H is equal to 3( H 1/3 -1), which is positive infinite. But the volume of the solid of revolution from 1 to H is equal to

3π(1 - H-1/3),

Which is finite and has standard part 3π.

We can give a simpler example of this phenomenon. Let H be a positive infinite hyperinteger, and form a cylinder of radius 1/H and length H² (Figure 6.7.6). Then the cylinder is formed by rotating a rectangle of length H², 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.

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

There are other types of improper integrals. If f is continuous on the half - open interval [a,b] then we define

_______ f(x)dx =________f(x)dx.

If f is continuous on (-∝,b] we define

_______ f(x)dx =________f(x)dx.

We have introduced four types of improper integrals corresponding to the four types of half - open intervals

[a,b), [a,∞), (a,b], ( - ∞,b].

By piecing together improper integrals of these four types we can assign an improper integral to most functions which arise in calculus.

DEFINITION

A function f is said to be piecewise continuous on an interval I if f is defined and continuous at all but perhaps finitely many points of I. In particular, every continuous function is piecewise continuous.

We can introduce the improper integral ____ f(x)dx whenever f is piecewise continuous on I and a,b are either the endpoints of I or the appropriate infinity symbol. A few examples will show how this can be done.

Let f 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.

EXAMPLE 7

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

∫ x-1/3 dx = ___ x2/3 + C.

Then

Similarly,

So

and the improper integral converges. Thus, the region shown in Figure6.7.7 has finite area.

Figure 6.7.7

If f is continuous on the open interval (a, b), the improper integral is defined as the sum

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

EXAMPLE 8 Find

The function 2/______ +1/____ is continuous on the open interval (0, 2)but discontinuous at both endpoints (Figure 6.7.8). Thus

Figure 6.7.8

First we find the indefinite integral.

Then

Also

Therefore

EXAMPLE 9 Find

The function 1/x² + 1/(x-1)² is continuous on the open interval (0,1) but discontinuous at both endpoints. The indefinite integral is

We have

Similarly we find that

In this situation we may write

and we say that the region under the curve in Figure 6.7.9 has infinite area.

Figure 6.7.9

Remark In Example 9

We are faced with a sum of two infinite limits. Using the rules for adding infinite hyperreal

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

If H and K are positive infinite hyperreal numbers and c is finite, then

H + K is positive infinite,

H + c is positive infinite,

-H - K is negative infinite,

-H + c is negative infinite,

H-K can be either finite, positive infinite, or negative infinite.

By analogy, we use the following rules for sums of two infinite limits or of a finite and an infinite limit. These rules tell us when such a sum can be considered to be positive or negative infinite. We use the infinity symbols as a convenient shorthand, keeping in mind that they are not even hyperreal numbers.

EXAMPLE 10 Find _____ xdx. We see that

and ______ xdx = ∞.

Thus ____ xdx diverges and has the form ∞-∞. We do not assign it any value or either of the symbols ∞ or -∞. The region under the curve f(x)=x is shown in Figure 6.7.10.

Figure 6.7.10

It is tempting to argue that the positive area to the right of the origin and the negative area to the left exactly cancel each other out so that the improper integral is zero. But this leads to a paradox.

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

Subtracting

But ______ 2dx=∞

So we do not give the integral_____ xdx the value 0, and instead leave it undefined.

PROBLEMS FOR SECTION 6.7

In Problems 1-36, test the improper integral for convergence and evaluate when possible.

1_________x -2 dx 2 ________x-0.9 dx

3________ x -1/2 dx 4______(2x - 1)-3dx
5______(2x - 1)-3dx 6_______ x -1/3 dx

7______x2 + 2x -1dx 8 _________x -2 - x -3 dx

9__________ x(1+x2)-2 dx 10 _______ x -1/2 + x-2 dx

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

13 _______(x-1) -2/3 dx 14 ________x-2 dx

15 _____x -2/3 dx 16 ______dx

17____ 2x(x2 -1) -1/3 dx 18____ 2x3 dx

19_____(2x-1) -2/3 dx 20_____(3x-1) -5 dx

21 ______x² dx 22 ______(2x -1 )3 dx

23_________dx 24_______x -1/3 dx

25 ________ x3 dx 26 _______x -3/2 dx

27________ dx 28______ |x| (x + 1) -3 dx

29_________dx 30_______(x-1 )-2 + (x-3) -2 dx

31____ (x-1)-1/2 + (3-x) -1/2 dx 32______dx

33__________ 34____________

35_________f(x)dx where f(x)= ______

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

37 Show that if r is a rational number, the improper integral ____ x-r dx converges when r <1

and diverges when r >1.

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

and diverges when r <1.

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

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

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

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

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

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

44 Find 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.

45 Find 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.

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

(b) the y-axis.

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

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

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

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

50 Do 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 the x-axis.

(b)Set up an integral for the area generated about the y-axis.

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

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

x-axis, (b) the y-axis (0< a ≤ r).

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

distance between them. If m1 is held fixed at the origin, find the work done in moving m2

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

55 Show that the Rectangle and Addition Properties hold for improper integrals.

EXTRA PROBLEMS FOR CHAPTER 6

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

square whose corners are on the original surface of the apple. If the original apple had radius r, find the

volume of the peeled apple.

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

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

on a line crossing the axis, one plane perpendicular to the axis and the other at a 45°angle.

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

intersects each plane perpendicular to the x-axis in a square.

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

5 y=0, y=________, 0 ≤ x ≤ 1

6 y=0, y=x3/2, 0 ≤ x ≤ 1

7 y= x, y= 4-x, 0 ≤ x ≤ 2

8 y=xp, y=xq, 0 ≤ x ≤ 1, where 0< q < p

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

the solid of revolution.

10 The 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.

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

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

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

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

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

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

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

A>0, B >0.

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

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

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

With respect to x.

21 Given that f(x) = xp, 0≤ x ≤ 1, p a positive constant, find a point c between 0 and 1 such that f(c)equals the average value of f(x)

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

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

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

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

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

positive constants.

27 A spring exerts a force of 10x 1bs when stretched a distance x beyond its natural length of 2ft. Find the work required to stretch the spring from a length of 3 ft to 4ft.

In Problems 28-36, test the improper integral for convergence and evaluate if it converges.

28 _________ x-3 dx 29 ____ (x+2) -1/4 dx

30 ____ x - 4 dx 31____ x -1/5 dx

32 ____ x 1/5 dx 33__________ dx

34 ________ dx 35 ________dx

36 _____ sin x dx

37 A wire has the shape of a curve y=f(x), a ≤x ≤b, and has density p(x) at value x.

Justify the formulas below for the mass and moments of the wire.

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

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

39 Find the mass, moments, and center of mass of a wire of constant density p bent in the

shape of the semicircle y=________ , -1 ≤x ≤1.

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

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

m= _____ p(x) π (f(x))² dx.

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

a ≤y ≤b, about the (vertical) y-axis.Water has constant density p per unit volume.

Justify the formula below for the amount of work needed to pump all the water to the top of the container.

W=______ pπ (g(y))² (b-y) dy.

42 Find the work needed to pump all the water to the top of a water-filled container in the shape of a cylinder

with height h and circular base of radius r.

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

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

height h, and circular top of radius r.

□45 The pressure, or force per unit area, exerted by water on the walls of a container is equal to

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

shape of a vertical rectangle of height b and width w, assuming the water comes to the top of the dam.

□46 A water-filled container has the shape of a solid formed by rotating the curve x=g(y), a ≤y ≤b about

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