阅读说明：本技术 Android系统微积分作业批改方法 (Method for correcting calculus operation of Android system ) 是由 伊斯拉木·吐尔逊 黄华 努尔古丽·艾力 阿布力米提·孜克力亚 王飞 李轮溟 于 2021-09-06 设计创作，主要内容包括：本发明公开了Android系统微积分作业批改方法,其方法步骤如下：S1：输入数学题；S2：按一定方式拆数学题；S3：建立数学题答案相关数据文件；S4：将数据文件放置网络服务器；S5：手机下载作业。在高校的高等数学授课过程中,给每一位学生布置不同题型的作业,学生做作业前,不用把题目写在演抄纸上,做题就在手机上完成,不像平时的完成作业需要很长一段时间并且抄袭,完成作业提交时,手机自动批改作业的同时每一道题的答案信息和打分情况发送到任课老师的手机,任课老师的手机按照学号存储这些信息以便以后查看,需要时,可以统计每种类型的题的得分率,可以参考学生们对这个知识点的掌握情况。(The invention discloses a method for correcting calculus operation of an Android system, which comprises the following steps: s1: inputting a mathematic question; s2: disassembling the mathematic questions according to a certain mode; s3: establishing a data file related to answers of the mathematic questions; s4: placing the data file in a network server; s5: and (5) downloading the operation by the mobile phone. In the course of higher mathematics teaching in colleges and universities, the homework with different question types is distributed for each student, before the students do homework, the students do homework without writing questions on the performance copy paper, and the homework is finished on the mobile phone, unlike the usual homework finishing which needs a long period of time and copy, when the homework is submitted, the mobile phone automatically corrects the homework and simultaneously sends the answer information and scoring condition of each question to the mobile phone of the teacher, the mobile phone of the teacher stores the information according to the number of the students for later viewing, and when necessary, the scoring rate of each type of questions can be counted, and the mastering condition of the students on the knowledge point can be referred.)

The method for correcting the calculus operation of the Android system is characterized by comprising the following steps: the method comprises the following steps: s1: inputting a mathematic question; s2: disassembling the mathematic questions according to a certain mode; s3: establishing a data file related to answers of the mathematic questions; s4: placing the data file in a network server; s5: and (5) downloading the operation by the mobile phone.

2. The Android system calculus job approval method of claim 1, wherein: in the step S1, a certain mathematical question and all possible answers thereof are input in Latex language, and during the input process, the part needing to be disassembled is currently inserted into,, ", andor "·" etc. the symbols,factors or characters before and after a symbol cannot swap positions, while factors or characters before and after a "·" symbol can swap positions.

3. The Android system calculus job approval method of claim 1, wherein: in the S2, Java program codes are used for disassembling an answer to a mathematical question written in a Latex format, the answer is a character array, and multiple nested item disassembling and positioning modes are adopted during disassembling the answer;

the problem and the answer of each operation problem are divided by a ═ sign, the calculation process of the mathematical problem is composed of a plurality of steps, each step is divided by the ═ sign, each step is composed of a plurality of items, each item in each step is divided by a group and a group, each page of operation contains an interference problem, and the operation problem and the interference problem are divided by a group and a group;

answer to tear down

Dividing the question and the answer by a ═ number;

dividing each step by a ═ number;

thirdly, using the 'number' to divide each item in the same step;

fourthly, use; (ii) a (ii) a "number to mark answers to the questions and distracters, add distracters to test students for too simple and appropriate purpose;

after the answers are disassembled according to the mode, the same characters are sorted and removed, the character data are stored in a four-dimensional character array of the caller item, and each member of the four-dimensional character array consists of a disassembled mathematical formula, a symbol and the serial number of the symbol on the current page;

when software is used for making questions, answers required by appropriate character spelling are selected from a plurality of options which are specified in advance, and the options which are split according to the signs of ═ and ',' contain characters with very high similarity to the standard answers, so that the options need to be further split;

one of the items to be unpacked consists of a number of factors, limit numbers, integral numbers, etc., and when unpacked, some items are unpacked in order, such as: the limit numbers, integral numbers, summation signs, etc. must be placed first, by taking the order apart, inserting them when entering the mathematical formulaNumber, unpack some items regardless of order, insert "·" number when entering;

new use ofThe fourth step of "split" stores the character string in item, while remembering the relative position of each character string before split, because the data obtained by this step for each item needing to be split twice is based on the Latex character string, the relative position of the character string in the fifth stepThe number division and the relative position before the "·" division, which totally include four pieces of information; the information is stored in item character againIn the cluster.

4. The Android system calculus job approval method of claim 1, wherein: in S3, an assistant program is used to display all the answers to each mathematical question, the assistant software is run to establish a class called "NOS", which has two members, one is an array of NOD type, the other is a character string, the NOD class also has two members, one is a character string, the other is an array of NOD type, the related items in the two steps are connected by drawing a straight line, and then the software automatically forms a related data, which indicates that some item in the next step is derived from some item in the previous step.

5. The Android system calculus job approval method of claim 1, wherein: in S4, the prepared data is placed at a location designated by the web server, and when the student clicks "get homework", the server randomly sends the homework to the designated mobile phone, thereby achieving the purpose of arranging different homeworks for different students.

6. The Android system calculus job approval method of claim 1, wherein: in the step S5, the software downloads job content, and stores the job content in the Android SQLite database, where the database is named "Task", and the field names are: ID, Student _ ID, HomeIndex, HomeItem, HomeText, Homeanswer, Time, Type;

the ID is a database record indexer;

student _ ID is a school number, and numbers in a character form are stored;

thirdly, each operation is divided into a plurality of pages, and the HomeIndex refers to the page of the operation;

the HomeItem refers to a page of the first question and stores numbers in a character form;

HomeText refers to the operation content, a mathematical formula in a Latex format and an array of judgment answers;

the HomeAnswer refers to an operation answer and stores a number with a symbol;

time refers to job layout Time, holding numbers in character form;

the Type indicates job, classroom, unit test, etc., and the digits are saved.

7. The Android system calculus job wholesale method of claim 3, wherein: in S2, each page of the software includes 3 to 5 (or more) mathematical questions, each of which may have 2 to 3 answers, each of which includes a plurality of steps, each of which includes a plurality of items, and a one-dimensional step score or other data is added, so that storing the information requires establishing a five-dimensional character array, a first dimension stores a certain mathematical question, a second dimension stores a certain practice of a certain question, a third dimension stores a certain practice of a certain question, a fourth dimension stores a certain item of a certain practice of a certain question, and a fifth dimension stores a symbol and a relative position of each item.

Technical Field

The invention relates to the technical field of mobile phone application information, in particular to a method for correcting calculus operation of an Android system.

Background

From agricultural civilization to industrial revolution to information society, the wave of digital technology has rolled the world, the propagation history of human civilization is reviewed, the great progress of information technology does not bring about huge change of life form each time, each major breakthrough does not push leap and promotion of human propagation quality again and again, along with the appearance of APP and the entering of mobile equipment into functional era, the development of APP also enters a new platform, the development and promotion of programming technologies such as JAVA and the like enable a plurality of application programs which can be freely installed/uninstalled by users to appear, meanwhile, the wireless network enables the connection of a mobile phone and the Internet to start generating APP of the Internet of things, so that the APP which is mainly entertainment at first starts to develop towards the directions of consultation, social contact, tools and the like, in the era of which the APP is mainly used as a carrier, the wireless network gradually infiltrates into the daily life field of people and plays the role of 'life assistant', providing convenience for daily life of people.

At present, university teaching is with the form textbook of closing a shift, because of the too many student population, in attendance, classroom mr and student interact, do homework/change homework etc. produce certain challenge, the main problem lies in: 1. in the aspect of homework making, the answer is plagiarized due to homework questions and tones; in addition, some students are somewhat lazy and are not willing to write; 2. in the aspect of teaching efficiency, the homework score and the peacetime score can not reflect the mastery condition of students on teaching key points by recording teaching records in a traditional mode, and meanwhile, teachers need to spend a great deal of time on correcting homework; 3. in a classroom, because the number of students is large, a teacher has no way to know the learning condition of each student, and the teacher has no way to check the whole process of exercise of the students due to time limitation. The problems are that the teaching efficiency is directly influenced, how to improve classroom interaction of students, how to keep various homework questions and reduce workload of teachers, and how to know the mastery condition of students on teaching key points according to homework and normal interaction conditions are a difficult problem to be solved urgently in the university teaching process at present, so that an Android system calculus homework correction method is proposed to be perfect.

Disclosure of Invention

The invention aims to provide a calculus operation correction method of an Android system to solve the problems in the background technology.

In order to achieve the purpose, the invention provides the following technical scheme:

the method for correcting the calculus operation of the Android system comprises the following steps: s1: inputting a mathematic question; s2: disassembling the mathematic questions according to a certain mode; s3: establishing a data file related to answers of the mathematic questions; s4: placing the data file in a network server; s5: and (5) downloading the operation by the mobile phone.

As a further scheme of the invention: in S1, a certain mathematical question and all possible answers are input in Latex language, and during the input process, the part to be disassembled is currently inserted with symbols such as "," ■ "or" · ", the factors or characters before and after the symbol" ■ "cannot be exchanged, while the factors or characters before and after the symbol" can be exchanged.

As a still further scheme of the invention: in the S2, Java program codes are used for disassembling an answer to a mathematical question written in a Latex format, the answer is a character array, and multiple nested item disassembling and positioning modes are adopted during disassembling the answer;

the problem and the answer of each operation problem are divided by a ═ sign, the calculation process of the mathematical problem is composed of a plurality of steps, each step is divided by the ═ sign, each step is composed of a plurality of items, each item in each step is divided by a group and a group, each page of operation contains an interference problem, and the operation problem and the interference problem are divided by a group and a group;

answer to tear down

Dividing the question and the answer by a ═ number;

dividing each step by a ═ number;

thirdly, using the 'number' to divide each item in the same step;

fourthly, use; (ii) a (ii) a "number to mark answers to the questions and distracters, add distracters to test students for too simple and appropriate purpose;

after the answers are disassembled according to the mode, the same characters are sorted and removed, the character data are stored in a four-dimensional character array of the caller item, and each member of the four-dimensional character array consists of a disassembled mathematical formula, a symbol and the serial number of the symbol on the current page;

when software is used for making questions, answers required by appropriate character spelling are selected from a plurality of options which are specified in advance, and the options which are split according to the signs of ═ and ',' contain characters with very high similarity to the standard answers, so that the options need to be further split;

one of the items to be unpacked consists of a number of factors, limit numbers, integral numbers, etc., and when unpacked, some items are unpacked in order, such as: the limit number, integral number, summation sign and the like must be placed at the first position, which adopts the steps of sequentially opening, inserting ' ■ ' when inputting a mathematical formula, and inserting ' (-) number when opening some items which are not related to the sequence;

sixthly, using ■ and- 'to further split the character string in item, while memorizing the relative position of each character string before splitting, because the data obtained by splitting each item twice in this step is divided by Latex character string, the relative position of the character string in the fifth stage, the relative positions before dividing according to ■ and the-' number, and includes four information in total. The information is restored in the item string.

As a still further scheme of the invention: in S3, an assistant program is used to display all the answers to each mathematical question, the assistant software is run to establish a class called "NOS", which has two members, one is an array of NOD type, the other is a character string, the NOD class also has two members, one is a character string, the other is an array of NOD type, the related items in the two steps are connected by drawing a straight line, and then the software automatically forms a related data, which indicates that some item in the next step is derived from some item in the previous step.

As a still further scheme of the invention: in S4, the prepared data is placed at a location designated by the web server, and when the student clicks "get homework", the server randomly sends the homework to the designated mobile phone, thereby achieving the purpose of arranging different homeworks for different students.

As a still further scheme of the invention: in the step S5, the software downloads job content, and stores the job content in the Android SQLite database, where the database is named "Task", and the field names are: ID, Student _ ID, HomeIndex, HomeItem, HomeText, Homeanswer, Time, Type;

the ID is a database record indexer;

student _ ID is a school number, and numbers in a character form are stored;

thirdly, each operation is divided into a plurality of pages, and the HomeIndex refers to the page of the operation;

the HomeItem refers to a page of the first question and stores numbers in a character form;

HomeText refers to the operation content, a mathematical formula in a Latex format and an array of judgment answers;

the HomeAnswer refers to an operation answer and stores a number with a symbol;

time refers to job layout Time, holding numbers in character form;

the Type refers to operation, classroom operation, unit test and the like, and the number is stored;

as a still further scheme of the invention: in S2, each page of the software includes 3 to 5 (or more) mathematical questions, each of which may have 2 to 3 answers, each of which includes a plurality of steps, each of which includes a plurality of items, and a one-dimensional step score or other data is added, so that storing the information requires establishing a five-dimensional character array, a first dimension stores a certain mathematical question, a second dimension stores a certain practice of a certain question, a third dimension stores a certain practice of a certain question, a fourth dimension stores a certain item of a certain practice of a certain question, and a fifth dimension stores a symbol and a relative position of each item.

Compared with the prior art, the invention has the beneficial effects that:

1. in the course of higher mathematics teaching in colleges and universities, each student is assigned with homework of different question types, before doing homework, the student does not need to write questions on the performance paper, but does the homework on the mobile phone, unlike the homework which needs a long period of time and can be copied at ordinary times, which is an unprecedented result, when the homework is submitted, the mobile phone automatically corrects the homework and simultaneously sends the answer information and scoring condition of each question to the mobile phone of the teacher, the mobile phone of the teacher stores the information according to the student number for later viewing, when needed, the score of each type of question can be counted, the mastering condition of the knowledge point by students can be referred, the teacher knows the condition of the students in real time and can take corresponding measures, through the application of the invention, the teaching effect and the learning enthusiasm of the students are greatly improved, and the teaching efficiency is improved, and simultaneously, the workload of teachers is reduced.

Drawings

Fig. 1 is a flow chart of a calculus work correcting method of an Android system.

Fig. 2 is a partial operation interface display diagram in the method for correcting the calculus operation in the Android system.

Fig. 3 is a partial operation interface display diagram in the method for correcting the calculus operation in the Android system.

Fig. 4 is a partial operation interface display diagram in the method for correcting the calculus job in the Android system.

Fig. 5 is a partial operation interface display diagram in the method for correcting the calculus job in the Android system.

Fig. 6 is a partial operation interface display diagram in the method for correcting the calculus job in the Android system.

Fig. 7 is a partial operation interface display diagram in the method for correcting the calculus job in the Android system.

Fig. 8 is a partial operation interface display diagram in the method for correcting the calculus job in the Android system.

Fig. 9 is a view showing a local operation interface in the method for verifying the calculus job in the Android system.

Fig. 10 is a view showing a user setting window in the method for verifying the calculus job in the Android system.

Fig. 11 is a view showing an acquired job window in the method for correcting the calculus job in the Android system.

Fig. 12 is a system setting window in the method for correcting the calculus job of the Android system.

Fig. 13 is an example of a display diagram of a calculus math problem displayed on an interface in the calculus operation approval method of the Android system.

Detailed Description

The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.

Referring to fig. 1 to 13, in an embodiment of the present invention, a method for correcting a calculus operation of an Android system includes the following steps: s1: inputting a mathematic question; s2: disassembling the mathematic questions according to a certain mode; s3: establishing a data file related to answers of the mathematic questions; s4: placing the data file in a network server; s5: and (5) downloading the operation by the mobile phone.

In S1, a certain mathematical question and all possible answers are input in Latex language, and during the input process, the part to be disassembled is currently inserted with symbols such as "," ■ "or" · ", the factors or characters before and after the symbol" ■ "cannot be exchanged, while the factors or characters before and after the symbol" can be exchanged.

In the S2, Java program codes are used for disassembling an answer to a mathematical question written in a Latex format, the answer is a character array, and multiple nested item disassembling and positioning modes are adopted during disassembling the answer;

the problem and the answer of each operation problem are divided by a ═ sign, the calculation process of the mathematical problem is composed of a plurality of steps, each step is divided by the ═ sign, each step is composed of a plurality of items, each item in each step is divided by a group and a group, each page of operation contains an interference problem, and the operation problem and the interference problem are divided by a group and a group;

answer to tear down

Dividing the question and the answer by a ═ number;

dividing each step by a ═ number;

thirdly, using the 'number' to divide each item in the same step;

fourthly, use; (ii) a (ii) a "number to mark answers to the questions and distracters, add distracters to test students for too simple and appropriate purpose;

each page of operation of the software consists of 3 to 5 (or more) mathematical questions, the answers of each mathematical question can be 2 to 3, each answer consists of a plurality of steps, each step consists of a plurality of items, one-dimensional storage step scores or other data are added, so that the information is stored by establishing a five-dimensional character array, a first dimension stores a certain mathematical question, a second dimension stores a certain practice of a certain question, a third dimension stores a certain step of a certain practice of a certain question, a fourth dimension stores a certain item of a certain step of a certain practice of a certain question, and a fifth dimension stores the symbol and the relative position of each item;

for example: string item ═ new String [ number of math questions per page ] [ several answers ] [ number of steps ];

after the answers are disassembled according to the mode, the same characters are sorted and removed, the character data are stored in a four-dimensional character array of the caller item, and each member of the thought character array consists of a disassembled mathematical formula, a symbol and the serial number of the symbol on the current page;

for example: \\ frac {1} { x }. sin3 x; + 37;

when the software is used for making questions, the answer required by appropriate character spelling is selected from a plurality of options specified in advance. The option of splitting according to the 'split', 'split' includes that the similarity of the characters and the standard answer is very large, so further splitting is needed;

one of the items to be unpacked consists of a number of factors, limit numbers, integral numbers, etc., and when unpacked, some items are unpacked in order, such as: the limit number, integral number, summation sign and the like must be placed at the first position, which adopts the steps of sequentially opening, inserting ' ■ ' when inputting a mathematical formula, and inserting ' (-) number when opening some items which are not related to the sequence;

sixthly, using ■ and- 'to further split the character string in item, while memorizing the relative position of each character string before splitting, because the data obtained by splitting each item twice in this step is divided by Latex character string, the relative position of the character string in the fifth stage, the relative positions before dividing according to ■ and the-' number, and includes four information in total. The information is stored in the item character string again;

for example: sin3 x; 37; 35_1_0:37_1_ 0; 35_2_0:37_2_ 1;

here 37 means sin3x originally in item 37; here 35_1_0:37_1_0 means that the second unraveling (i.e., unraveling by the "") occurs twice, occurs at items 35, 37, and the relative position is 1, unravels into one item; 35_2_0:37_2_1 means that the third split occurs twice, and in items 35 and 37, the split is 2 items, and the relative positions are 0 and 1, respectively.

In S3, when the student makes questions and clicks on each item, the software determines, according to the location information, that the clicked item and the previous clicked item may not be pieced together into the original string, because some items in the two steps before and after some answers to the mathematical questions remain unchanged, and the answer in the last step of some mathematical questions consists of several items. In order to solve the problem, an auxiliary program is used for displaying all answers of each mathematical question, the auxiliary software establishes a class called NOS when in operation, the class has two members, one is an array of NOD type, the other is a character string, the NOD class also has two members, one is a character string, the other is an array of NOD type, related items in the two steps before and after are connected in a straight line drawing mode, and then the software automatically forms related data which indicates that a certain item in the next step is derived from a certain item in the previous step;

for example, (sin3xlnx) ' (sin3x) '. lnx, + sin3x (lnx) ' -cos 3 x- (3x) '. lnx, + frec {1} { x }. sin3x +' cos3 x.3. lnx, + frec {1} { x }. sin3x < > { +33{ +35{ +37} } } { +34{ +36}, where +37 is derived from +35 and +35 is derived from + 33; here +36 is pushed from + 34.

In S4, the prepared data is placed at a position designated by the web server, and when the student clicks "get homework", the server randomly sends the homework to the designated mobile phone, so as to achieve the purpose of arranging different homeworks for different students, which is the final purpose of compiling the software.

In the step S5, the software downloads job content, and stores the job content in the Android SQLite database, where the database is named "Task", and the field names are: ID, Student _ ID, HomeIndex, HomeItem, HomeText, Homeanswer, Time, Type;

the ID is a database record indexer;

student _ ID is a school number, and numbers in a character form are stored;

thirdly, each operation is divided into a plurality of pages, and the HomeIndex refers to the page of the operation;

the HomeItem refers to a page of the first question and stores numbers in a character form;

HomeText refers to the operation content, a mathematical formula in a Latex format and an array of judgment answers;

the HomeAnswer refers to an operation answer and stores a number with a symbol;

time refers to job layout Time, holding numbers in character form;

the Type refers to operation, classroom operation, unit test and the like, and the number is stored;

the software displays a plurality of pieces of information on a homepage of the software according to the content of the database and the number of HomeIndex, wherein each piece of information occupies one line, and each line consists of the time of job arrangement and the time of job downloading;

when a user clicks a certain line, corresponding operations are displayed, each operation comprises 3-5 (or more) mathematic questions, at the moment, the operation content is completely disassembled once again according to the method, the operation content is divided into two parts, an operation title and answer items, an operation page is composed of an upper part and a lower part, the title is displayed on the upper part, and a plurality of items are displayed on the lower part and comprise equal signs, plus, minus and interference items;

before displaying the operation, the software establishes a two-dimensional integer array according to the number of questions contained in the current page, the array stores the number of the questions clicked by the user and the number of options, and in the process of making the questions, the software stores the data corresponding to each click of the user in a HomeAnswer field in a database;

when the user completes the job click "submit job":

firstly, the software opens a database and extracts answers from the HomeAnswer;

secondly, the software runs getCombine and build functions, and then positioning data contained in the item array is used for judging whether spliced options of the user are reasonable or not, namely whether positioning information of the disassembled item is consistent with position information of the items in front and back, which is the first step of judgment, if all items selected by the user are compatible, the program executes to arrange user answers and can do the next step;

thirdly, software establishes a three-dimensional NOD type array, extracts the associated data of HomeText, and constructs NOD type members according to the associated data;

comparing the answer sorted in the step two with the data of the NOD array by the software, and mainly checking whether each step contains corresponding items and items which do not appear at the time and whether the step number reaches a preset number;

judging that the question generally has only one step, so that the written answer is given full mark, if the standard answer of the calculated question has two steps, each step takes half of the mark, if the standard answer of the calculated question has more than three steps, the three steps including the last answer are given full mark.

Software main interface

An operation interface of software in the mobile phone is shown in fig. 2: the screen displays the class, the school number, the name and the homework list of the students, wherein '1000102020-04-08, 1000112020-04-08 and …' are homeworks obtained by classmates with the school number of 220132087, and the screen displays the list by clicking to open '1000102', as shown in fig. 3: there are A, B, C, D, E, F questions, when the student clicks any one question, the color of the question changes from black to blue, and the equal sign is displayed below, and clicking the equal sign starts to do the question, for example, finding the derivative of ln3x, clicking the equal sign first, and then clicking in the following optionsThe following is shown in fig. 4, and then click g.3, so as to spell the answer of the question in one step, the answer of the question is: as shown in fig. 5, when each question is made, the answer of all the questions on the page is displayed as follows according to the method, as shown in fig. 6, the user makes a mistake, can click "cancel", finish one page, press "next", can continue to finish the next page of job, finish the current job and click "return" as shown in fig. 7, return to the main window, click "submit job" as shown in fig. 8, display a next window, as shown in fig. 9, input the formulated IP address, then determine, at this time, the software automatically judges whether each step of each question is correct, remove the repeated answer, and score the questions one step by one step, wherein the scoring criteria is: one question and one score are taken, the last answer is written to obtain 0.5 score, at least one step is written to obtain 0.5 score except the last step, the distribution obtained by the homework is finally sent to the server, the user sets a window, as shown in figure 10, the class, the name and the school number of the student can be modified in the window, the homework can be deleted, the homework obtaining window is shown in figure 11, and the system sets a window, as shown in figure 12.

Example job question format:

1.(22 min) D { (x, y) |0 ≦ x ≦ 1, and 0 ≦ y ≦ 1}, then

2.(22 min) D is a two-dimensional coordinate axis and a straight line x + y

When D is X-type, the closed region is defined by 2

(22 min) D is a triangular closed region with vertices (0,0), (pi, 0) and (pi, pi), respectively, and is considered to be X-type,

(22 min) D is defined by the line y-2, y-x and y

2x, and D is a Y type,

(12 min) exchanging integral order-₀ ¹dy∫₀ ^yf(x,y)dx＝.∫₀ ¹dx∫_x ¹f(x,y)dy

The following character string is obtained through processing:

1 (22) D ═ left \ left { \ left (X, y \ right) |0\ leqx \ leq1,0\ leqy \ leq1\ right \ then \ iint _ Dxd \ sigma ═ int _0^ 1. xdx ^1dy ^ fr \ {1} {2}, left [ X ^2\ right ] _0^ 1. RTM.1. left ^ left [ X ^2\ right ] _ 1} fr {1} {2}, 1 ^ left ^ 1} D ^2\ right } D ^2 \\\\\ \ y \ \ 2\ \ y \ \ 1 ^ D ^2 ^ X ^2\ \ y \ \ 2\ \ y \ \ 2\ \ y \ coordinate axis (22) D + 2\ coordinate axis is defined by two coordinate axes of two coordinate axes (X and 2\ \ int 2 ^2\ \ int 2 ^2\ \ y \ \ 2\ \ y \ \ 2\ \ y \ \ 3\ \ y \ \ 3\ \ y \ \ 2\ \ y \ \ 2\ \ y \ \ 3\ \ y \ \ 3\ \ y \ \ 3\ \ y \ \, - \\ frac {2} {3} -/left [ X ^3\ right ] _0^2, - +4 · \ left [ X \ right ] _0^2 \\ \ front {20} {3} \ \3.(22 points) D is a triangular closed region with vertices of 0,0, \\\ pi,0 and \ pi, \\ pi, D is of the X type, and \\ ixcoos \ left (X + y right) D \\ \ int _0^ X \ \ dy \ left (X + y \ pi \ \ X \ \ y \ \ left (X + y \ pi), and \ \ pi \ \ 2\ \ y \ \ right) is a { 0^ 0\ \ y \ \ right \ \ 2\ \ y \ \ right \ \ 2\ \ int \ \ y \ \ right \ \ 2\ \ y \ \ sin \ \ 2\ \ y \ \ left \ \ y \ \ right \ \ y \ \ left (X \), y ═ x and Y ^2x enclose a closed area, D is Y type, \\ int _ D (x ^2+ Y ^2-x) D \ sigma \ int _0^2dy \ int \ frac { Y } {2 ^ Y (x ^2+ Y ^2-x) dx ^ int _0^2\ left [ \\ \ frac { x ^3} {3} + Y ^2x- \\ \ frac { x ^2} {2} \\\\\ right ] _ frac {2} { ydy \ Y {19} {24} { 0^2 ^ Y {3},y, - \\ frac {3} {8} · \ int _0^2y ^2dy ═ frac {13} {6} \ \5.(12 min) exchange the integration sequence \ int _0^1dy \ int _0^ yf \ left (x, y \ right) dx ^ int _0^1dx \ int _ x ^1f (x, y) dy < > { +13{ +14{ +15{ +16} } } }; (ii) a { +17{ +18{ +19{ +20{ +23} } { -21{ +23} } { +22{ +23} } } } } } } } } }; (ii) a { +24{ +25{ +26{ -27{ -29} } { +28{ -29} } } } } } } } }; (ii) a { +30{ +31{ +32{ +34} } { -33{ +34} } } } } }; (ii) a {+35}

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Although the present invention has been described in detail with reference to the foregoing embodiments, it will be apparent to those skilled in the art that various changes in the embodiments and/or modifications of the invention can be made, and equivalents and modifications of some features of the invention can be made without departing from the spirit and scope of the invention.