The Teaching Design of Triple Integral and Its Application in College Students' Mathematics Competition

: Triple integral is a difficult point in the teaching of higher mathematics. How to teach a triple integral course and learn to apply it is particularly important. Through the teaching content design of triple integral and its application in college students ' mathematics competition, this paper aims to make students better understand the content through clear teaching framework and explanation ideas, simple and easy to understand teaching methods and concise expression methods, which has certain reference value for teachers ' teaching, guiding students to participate in the competition and participating students.


Introduction
Triple integral is an important concept in higher mathematics and a method of integrating functions in a certain area of three-dimensional space.Triple integral in rectangular coordinate system is a very important tool, which can be used to solve many practical problems, such as physics, engineering, computer graphics and other fields.Through in-depth study and understanding of triple integral, we can better grasp the nature of functions in space and provide more effective mathematical tools for solving practical problems.The triple integral is a difficult knowledge point in the double integral, especially the calculation of the triple integral, many students generally find it difficult to do the problem, based on this, this paper combs the teaching content of the triple integral, and combines the test points of the triple integral in the national college students ' mathematics competition over the years, makes a general summary, and lists the knowledge points according to the teaching content, which is concise, clear and easy to understand.

Teaching Design Ideas
Firstly, the concept of triple integral is introduced through an example, and then the nature and geometric significance of triple integral are briefly introduced.Then, the calculation of triple integral is mainly introduced, and the specific conversion formulas in rectangular coordinate system, cylindrical coordinate system and spherical coordinate system are introduced.Finally, the content summary, classroom practice, homework assignment and classroom extension are carried out.The teaching content frame is shown in Figure 1 [1][2][3][4][5][6].

Teaching Content Design
In order to effectively use the limited time of the classroom and better introduce the content of the triple integral, the teaching content is summarized below, and the classroom teaching of the triple integral is realized according to the design scheme.

Example
The mass of uniformly distributed object: M = volume density×volume (volume density is constant) The non-uniform distribution of the object, known volume density ) , , ( z y x  , find its mass M. (volume density is a function) The solution: ' large into small, constant generation, approximate and, limit ' (1) Segmentation: the object is divided into n pieces of

Let
) , , ( z y x f be a bounded function on the space bounded closed region  .Segmentation: The closed region  is arbitrarily divided into n small closed regions


, where represents the i th small region and its volume.
Approximate summation: take a point i v  on each ) , , ( , and make and is the maximum diameter of each small closed region) If the limit of the sum exists, the limit is called the triple integral of the function in the rectangular coordinate system, and the triple integral can be expressed as in the rectangular coordinate system.

Properties of Triple Integrals
The property of triple integral is similar to that of double integral, and it is no longer repeated.Only one property of simplifying the calculation of triple integral is introduced.
If the integral region  is symmetric with respect to the plane 0  x , then Similarly, if the integral region  is symmetrical about the plane 0  y or 0  z , there is a similar conclusion.

Geometric Meaning of Triple Integral
The mass (1) Rectangular coordinate system Method 1: Projection method (one before two) By projecting the closed region  to the xOy plane, a plane region xy D is obtained, and the boundary of xy D is used as the reference line to make the cylindrical surface parallel to the z axis.This cylinder divides the boundary surface S of  into two parts, ) , ( : Method 2: Cross-section method (first two then one).
Assume that the region  is located between two planes a z  and b z  ) ( b a  perpendicular to the z axis, the cross section perpendicular to the x axis is denoted as z D , and the cross section is projected onto the xOy plane, then the closed region  can be expressed as , and the triple integral can be transformed into When the integrand contains only z functions, then Applicable conditions: (1) the integrand contains only the function of z ; (2) The section z D perpendicular to the z axis should be regular and easy to calculate the area.
  Three sets of coordinate surfaces, when r is a constant, represent the cylindrical surface; when  is a constant, it represents the half plane; when z is a constant, it represents the plane.
The volume element dv in the cylindrical coordinates is surrounded by six coordinate planes:  (1) Calculation of triple integral, select the coordinate system is the key, from the characteristics of the integrand and integral region to consider: 1) When the integral region  is cuboid, tetrahedron or arbitrary shape, choose the rectangular coordinate system → projection method (first then two).
2) When the integrand function is , choose the rectangular coordinate system → cross section method (first two then one); 3) When the integral region  is a rotating conical surface, a rotating parabolic surface, and a spherical surface, the cylindrical coordinate system is selected; 4) When the integral region  is spherical, the spherical coordinate system is selected. .Tip: Select the rectangular coordinate system, method one: projection method (first one after two); method 2: Cross-section method (first two then one).
2) Calculate the triple integral , where  is the cube surrounded by a cone

After-school Homework
Textbook after-school exercises

Knowledge Extension
In order to stimulate students ' interest in learning mathematics and cultivate students ' ability to analyze and solve problems, after teaching the content of triple integral, this paper gives several typical real problems of college students ' mathematics competition over the years, expands students ' knowledge, and exercises their mathematical thinking and problem solving ability.The specific topics are as follows.
1) (2016 Eighth National College Students ' Mathematics Competition Preliminary Questions) The space area where an object is located is , the density function is , and the mass is calculated.Analysis: Steps of triple integral calculation: Step 1: Draw the figure of the integral area, investigate whether the integral area has symmetry, the symmetry and rotation symmetry of the coordinate surface, the non-standard integral area, and the conversion to the standard integral area.
Step 2: Determine the coordinate system and construct the cubic definite integral expression Step 3: Calculate three definite integrals The center point On the symmetry of three coordinate planes, it has rotational symmetry.


The spherical coordinate calculation method of triple integral:

2) (2017 Ninth National College Students ' Mathematics Competition Preliminary Competition)
Let the space region surrounded by the surface Step 3: Determine the coordinate system of the conversion of repeated integration Step 4: Construct the expression of cumulative integral.
Step 5: Calculate three definite integrals Four methods: Spherical coordinate system Cylindrical coordinate system Rectangular space coordinate system (projection method ' first one after two ') Spatial rectangular coordinate system (section method ' first two then one ') Method 1: spherical coordinate system Answer: Method 3: space rectangular coordinate system (projection method' first one after two ') Method 4: space rectangular coordinate system (section method ' first two after one ') Answer: is calculated, where  is the area surrounded by the in the first quadrant.Analysis: The first step is to determine and construct the coordinate system for calculating the triple integral.By analyzing the graphic features, it can be known from the surface expression that the spherical coordinate system should be established. is the angle between the projection line ON of the positive half axis of x rotating counterclockwise on the xOy coordinate plane to the point on the surface and the connection line OP of origin on the xOy coordinate plane. is the angle between the vector OP and the positive direction of the z axis from the origin to the point on the surface; r is the distance from the point on the surface to the origin.
Ball coordinates: The second step: the boundary surface equation and the integrand function are used to describe the surface equation with the corresponding coordinate variables, and the spherical coordinates are substituted into the surface Step 3: Determine the scope of the integral variable and construct the expression of the cumulative integral.
Step 4: Calculate the cumulative integral to get the integral result. Answer:

Conclusion
By summarizing the teaching design of triple integral and its application in college students ' mathematics competition, it shows a clear teaching framework and explanation ideas, simple and easy to understand teaching methods, concise and comprehensive expression, so that students can better grasp the content of triple integral, especially the calculation methods and skills of triple integral [13][14][15].At the same time, it is also helpful to improve students ' ability to analyze and solve logical thinking, stimulate students ' interest in learning, improve teachers ' classroom teaching effect, and have certain reference value for teachers ' teaching, guiding students to participate in competitions and participating students [16][17].

Figure 1 :
Figure 1: Teaching Content Frame Diagram density in the space finite closed region  is given.

Example 2
Calculates the triple integral   zdxdydz , where  is a closed region surrounded by method (first two then one).a point in space, and let the polar coordinate of the projection P of point M on the xOy plane be  , r , then such three numbers z r , , are called the cylindrical coordinates of point Mof rectangular coordinates to cylindrical coordinates, where (1) the half plane with the angle of  and   d  ; (2) Cylindrical surfaces with radius r and dr r  ;(3) Planes of height z and dz z  .So the triple integral calculation formula is in space, then point M can be determined by three ordered numbers, where r is the distance between the origin O and point M,  is the angle between the directional line segment OM and the positive axis, and  is the angle from the positive z axis meter to the directional line segment OP in the counterclockwise direction, where P is the projection of point M on the xOy plane.Such three numbers Three sets of coordinate surfaces, when r is a constant, represent the sphere; when  is a constant, it represents a cone; when  is a constant, it represents the half plane.The volume element dv in spherical coordinates is surrounded by six coordinate surfaces: (1) half plane  and   d  ; (2) Conical surface  and   d  ; (3) Spheres with radius r and dr r  .So the triple integral calculation formula is: spherical coordinate system to calculate triple integral: the spherical coordinate system.
general steps of triple integral calculation: Step 1: Draw the graphics of the integral region Step 2: Simplify and transform the integral model by means of symmetry.
3. (In 2019, the 11th National College Students ' Mathematics Competition Preliminary Questions) equation does not mention the value of z , so 2

Table 1 :
Summary of Calculation of Triple Integral