Multiple integral

Japanese: 重積分 - じゅうせきぶん

A method for integrating functions of multiple variables. Below, we explain the case of two variables. Consider a rectangle R on the xy plane with sides parallel to the coordinate axes, as shown in Figure A.

R : a ≦ x ≦ b , c ≦ y ≦ d
The area of such a rectangle, ( b - a )( d - c ), will be represented as | R | below. Now, for a figure G that is included in R , let us determine the area of G. Now, arbitrarily subdivide the interval [ a , b ] [ c , d ] and set the abscissas as follows, in order.

a = x ₀ < x ₁ < x ₂ <...< x _n = b
c = y ₀ < y ₁ < y ₂ <...< y _m = d
This divides R into nm small rectangles, and of these small rectangles, let R ₁ , R ₂ ,……, R _p be those that are completely contained in G , and let R 1 , R 2 ,……, R _p , R _p +1 ,……, R _p _{+ q} be those that share at least one point with G. When one partition of R is defined _, | R ₁ | + | R ₂ | + … + | R _p | is determined, and the upper limit of this value obtained by changing the partition in various ways is called the interior area of G , or the Jordan interior measure, or the Jordan content. In other words, this is the limit value of | R ₁ | + | R ₂ | + … + | R _p | when the partition size is made finer. Similarly, the lower limit of the value of _| R1 |+ R2 |+......+| Rp _| +| Rp _{+ 1} |+......+| Rp _{+ q} | determined for _each partition when the partition is changed in various ways is called the exterior area of G , Jordan exterior measure, Jordan exterior capacity, etc. When the exterior area and interior area are the same, G is said to have a definite area, and this common value is called the area of G.

Let G be a set of definite areas contained in a rectangle on a plane, and the meaning of integrating a bounded function f ( x , y ) on this set is defined as follows. That is, divide R into smaller rectangles, R ₁ , R ₂ ,……, R _p , that are contained in G , and define ( a _k , b _k ) as a point in R _k , and sum

This is nothing but the volume of the approximate figure when the figure enclosed by the graph of f and the xy plane is approximated by a group of small rectangular parallelepipeds with base area | R _k | and height f ( a _k , b _k ). When the division is changed in various ways and the way the points ( a _k , b _k ) are taken within each small rectangle is changed in various ways, if the size of the division is reduced in any way, at the limit, it always approaches a constant value A , then,

This is called the double integral of f ( x , y ) on G. Or, simply, the integral of f ( x , y ) on G. The right-hand side symbolically represents the infinite sum of the volume ( height x base area = f ( x , y ) dxdy ) of a rectangular parallelepiped of height f ( x , y ) with a base of an infinitesimal rectangle of width dx and height dy. When an integral of f ( x , y ) on G exists, f ( x , y ) is said to be integrable on G. Bounded continuous functions are always integrable. The usual definite integral of one variable represents the area between the graph of y = f ( x ) and the x- axis. Similarly, it should be intuitively clear from the above definition that the multiple integral represents the volume of a columnar part erected on G between the surface obtained as the graph of z = f ( x , y ) and the xy plane.

[Example 1] Let f ( x , y ) = x . Let G be inside the triangle that connects the origin, the point x = a on the x- axis, and the point y = a on the y- axis.

gives the volume of the prism in Figure B (the calculation method will be shown later).

[Example 2]

Let G be the interior of the circle x ² + y ² = a ² in the xy plane. Then, the definite integral

is the volume of the hemisphere ( Figure C ).

[Osamu Takenouchi]

Repeated Integration

Although the multiple integral has been defined in this way, it is difficult to find its value as is, so we will transform it into a repeated integral of one variable. Now, let's assume that the range of integration G is a figure surrounded by two straight lines x = a , x = b and curves y = ₁ ( x ), y = ₂ ( x ) by the continuous functions 1( x ), ₂ ( x ) ( ₁ ( x ) < ₂ ( x )) defined in the interval [a , _b ] ( Figure D ). Then,

As shown above, the value of a multiple integral can be found by repeatedly integrating one variable.

In the case of [Example 1], G is a triangle bounded by 0≦ x ≦ a , y ＝0 and y ＝ a － x . Therefore,

In the case of [Example 2], G is - a ≦ x ≦ a ,

It is a circle surrounded by . Therefore,

The integral on the inside is the radius

Since this represents the area of the upper semicircle of a, its value is (π/2)( a ² － x ² ), so this integral is

[Osamu Takenouchi]

Change of integration order

There are two ways to make a multiple integral into a repeated integral: integrate with respect to y and then integrate with respect to x , or integrate with respect to x and then integrate with respect to y . By using this inversely, when a repeated integral is given first, the order of the integrals can be changed. Let me explain with an example.

In this case, it is possible to find the integral with respect to x inside, but it will be a very complicated process. Therefore, we change the order of the integrals and perform the integral with respect to y first. In this case, the integral range G is shaped as shown in Figure E. If we pay attention to this,

Therefore, it is relatively easy to obtain.

[Osamu Takenouchi]

Variable transformation in multiple integrals

When changing integral variables in multiple integrals, use the Jacobian as follows. If you change the variables from x and y to u and v by x = ( u , v ), y = ψ( u , v ), and the range of integration G changes to H on the uv plane, then

where

is the Jacobian.

In [Example 2], if we convert to polar coordinates x = r cosθ, y = r sinθ, H becomes 0≦ r ≦ a , 0≦θ≦2π,

Therefore, we have the following:

The above method for multiple integrals is the Riemann integral approach. There are many difficulties in applying this method to general theory. If you use Lebesgue integrals, you can get a very clear answer. In Lebesgue integrals, the relationship between multiple integrals and repeated integrals is called Fubini's theorem.

[Osamu Takenouchi]

[Reference] | Riemann integral | Lebesgue integral

Multiple integral explanation diagram (Figure A)
©Shogakukan ">

Multiple integral explanation diagram (Figure A)

Multiple integral (Example 1) [Figure B]
©Shogakukan ">

Multiple integral (Example 1) [Figure B]

Multiple integral (Example 2) [Figure C]
©Shogakukan ">

Multiple integral (Example 2) [Figure C]

Repeated integral (Figure D)
©Shogakukan ">

Repeated integral (Figure D)

Multiple integrals (changing the order of integration) [Figure E]
©Shogakukan ">

Multiple integrals (changing the order of integration) [Figure E]

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

多変数の関数を積分する方法。以下、2変数の場合について説明する。図Aのようにxy平面上の、座標軸に平行な辺をもつ長方形Rを考える。

　　R : a≦x≦b, c≦y≦d
このような長方形の面積(b－a)(d－c)を以下、|R|で表す。そして、Rに含まれるような図形Gについて、Gの面積を定めよう。いま、区間［a, b］［c, d］を任意に細分して、その分点を順に次のようにする。

　　a＝x₀＜x₁＜x₂＜……＜x_n＝b
　　c＝y₀＜y₁＜y₂＜……＜y_m＝d
これによって、Rはnm個の小長方形に分割されるが、これらの小長方形のうち、Gにすっかり含まれてしまうものをR₁, R₂,……, R_pとし、Gと少なくとも1点を共有するものをR₁, R₂,……, R_p, R_p+1,……, R_p+qとする。Rの分割を一つ定めると、|R₁|＋|R₂|＋……＋|R_p|が定まるが、分割をいろいろ変えて得られるこの値の上限を、Gの内側の面積、あるいはジョルダンの内測度、またはジョルダンの内容量という。これはすなわち、分割の大きさを細かくしていったときの、|R₁|＋|R₂|＋……＋|R_p|の極限値である。同様に、各分割において定まる|R₁|＋R₂|＋……＋|R_p|＋|R_p+1|＋……＋|R_p+q|の値の、分割をいろいろ変えたときの下限を、Gの外側の面積、ジョルダンの外測度、ジョルダンの外容量などとよぶ。外側の面積と内側の面積が一致するとき、Gは面積確定であるといい、この共通の値をGの面積という。

　Gは平面上のある長方形に含まれた面積確定な集合であるとし、この集合の上で、有界な関数f(x, y)を積分することの意味を次のように定める。すなわち、Rを細分し、その分割の小長方形のうちGに含まれるものR₁, R₂,……, R_pについて、(a_k, b_k)をR_k内の1点として、和

をつくる。これはfのグラフとxy平面で囲まれる図形を、底面積|R_k|、高さf(a_k, b_k)の小直方体の集まりで近似したときの、近似図形の体積にほかならない。分割をいろいろに変え、また、各小長方形内で点(a_k, b_k)のとり方をいろいろに変えるとき、どのようにしても、分割の大きさを小さくしていくと、その極限において、つねに一定の値Aに近づくならば、

と表し、これをf(x, y)のG上の二重積分という。または単にf(x, y)のG上の積分という。右辺は象徴的に、横dx、縦dyなる微小長方形を底面にもつ、高さf(x, y)の直方体の体積（高さ×底面積＝f(x, y)dxdy）の無限和をとることを表している。f(x, y)のG上の積分が存在するとき、f(x, y)はG上で積分可能であるという。有界連続関数は、つねに積分可能である。通常の1変数の定積分は、y＝f(x)のグラフとx軸との間に挟まれた部分の面積を表すものであった。同様に、重積分は、z＝f(x, y)のグラフとして得られる曲面とxy平面との間の、Gの上に立てた柱状部分の体積を表すことは、以上の定義により直観的には明らかであろう。

〔例1〕f(x, y)＝xとし、Gを原点とx軸上x＝aの点、およびy軸上y＝aの点を結ぶ三角形の内部とするとき、

は、図Bのプリズム形の体積を与える（計算の方法については後に示す）。

〔例2〕

とし、Gをxy平面内の円x²＋y²＝a²の内部とすれば、定積分

は半球の体積である（図C）。

［竹之内脩］

繰り返し積分

重積分はこのように定義されたが、その値を求めるのには、そのままでは困難であるので、1変数の積分の繰り返しに変形する。いま、積分の範囲Gが、区間［a, b］で定義された連続な関数₁(x), ₂(x)（₁(x)＜₂(x)とする）によって、2直線x＝a, x＝bと、曲線y＝₁(x), y＝₂(x)によって囲まれた図形であるとする（図D）。そうすると、

のように、重積分の値は1変数の積分を繰り返すことによって求められる。