黎曼积分与菲氏积分

回顾过去，上世纪60年，苏联菲氏微积分教材采用的积分定义是十九世纪高斯弟子黎曼给出的积分定义，国内照搬不误。

    进入本世纪，塔尔斯基弟子Keisler推出内容全面覆盖菲氏微积分教科书的非准微积分教材，简明易懂，堪称世界一流。

  本文明附件是非标准黎曼积分的内容，图形省略。   

袁萌   陈启清   1月26日

附件:Limann’s INTEGRATION

.1 THE DEFINITE INTEGRAL

We shall begin our study ofthe integral calculus in the same way in which we began

with the differential calculus-by asking a question about curves in the plane.

Suppose f is a real function continuous on an interval I and consider the

curve y = f(x). Let a < b where a, bare two points in J, and let the curve be above the

x-axis for x between a and b; that is, f(x) ~ 0. We then ask: What is meant by the

area of the region bounded by the curve y = f(x), the x-axis, and the lines x = a and

x = b? That is, what is meant by the area of the shaded region in Figure 4. 1.1? We

call this region the region under the curve y = f(x) between a and b.

y

a b X

Figure 4.1 .1 The Region under a Curve

The simplest possible case is where f is a constant function; that is, the curve

is a horizontal line f(x) = k, where k is a constant and k ~ 0, shown in Figure 4.1.2.

In this case the region under the curve is just a rectangle with height k and width

b - a, so the area is defined as

Area= k·(b- a).

The areas of certain other simple regions, such as triangles, trapezoids, and semicircles,

are given by formulas from plane geometry.

175

176 4 INTEGRATION

y

f(X) = k

0 a b X

area = k(b - a)

Figure 4.1.2

The area under any continuous curve y = f(x) will be given by the definite

integral, which is written

fj(x)dx.

Before plunging into the detailed definition of the integral, we outline the main ideas.

First, the region under the curve is divided into infinitely many vertical

strips of infinitesimal width dx. Next, each vertical strip is replaced by a vertical

rectangle of height f (x ), base dx, and area j (x) dx. The next step is to form the sum

of the areas of all these rectangles, called the infinite Riemann sum (look ahead to

Figures 4.1.3 and 4.1.11). Finally, the integral J~ f(x) dx is defined as the standard

part of the infinite Riemann sum.

The infinite Riemann sum, being a sum of rectangles, has an infinitesimal

error. This error is removed by taking the standard part to form the integral.

It is often difficult to compute an infinite Riemann sum, since it is a sum of

infinitely many infinitesimal rectangles. We shall first study finite Riemann sums,

which can easily be computed on a hand calculator.

Suppose we slice the region under the curve between a and b into thin vertical

strips of equal width. If there are n slices, each slice will have width Llx = (b - a)jn.

The interval [a, b] will be partitioned into n subintervals

[x0 , x 1], [x 1 , x2], ••• , [x11 _ 1 , X 11],

where x0 = a,x 1 =a+ Llx,x2 =a+ 2Llx, .. . ,X11 =b.

The points x0 , x 1 , ... , X 11 are called partition points. On each subinterval [xk _ 1 , xk],

we form the rectangle of height f(xk- d. The kth rectangle will have area

From Figure 4.1.3, we can see that the sum of the areas of all these rectangles will be

fairly close to the area under the curve. This sum is called a Riemann sum and is equal

to

f(x 0 ) Llx + f(x 1) Llx + · · · + /(x,_ 1) Llx.

It is the area of the shaded region in the picture. A convenient way of writing Riemann

sums is the "l:-notation" (l: is the capital Greek letter sigma),

h I f(x) Llx = f(x0 ) Llx + /(x 1) Llx + · · · + /(x11 _ 1) Llx.

4.1 THE DEFINITE INTEGRAL 177

f(x)

x6 x 7 = b X

Figure 4.1.3 The Riemann Sum

The a and b indicate that the first subinterval begins at a and the last subinterval ends

at b.

We can carry out the same process even when the subinterval length ~x does

not divide evenly into the interval length b- a. But then, as Figure 4.1.4 shows, there

will be a remainder left over at the end of the interval [a, b], and the Riemann sum will

have an extra rectangle whose width is this remainder. We let n be the largest integer

such that

a+ n ~x.::::; b,

and we consider the subintervals

[xo, xJl, ... , [xn-1, xn], [xll, b],

where the partition points are

x 0 = a, x 1 = a + ~x, x 2 = a + 2 ~x, ... , x" = a + 11 ~x, b.

f(x)