Applications of Integration

Applications of Integration

In my previous posts, we discussed Definite and Indefinite Integrations. Now we shall learn about Applications of Derivatives. Initially, we shall discuss “Area Under Curves”.

Area Under Curve-: If we want to calculate the area between the curves y=f(x) and y=g(x) then there are actually two cases-

First Case when   f(x) \ge g(x)Below is the figure showing this case


here area under these  two curves       


The second Case When  f(x) \le g(x)Below figure shows this case


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Definite Integration-Topics in IB Mathematics

Definite Integration

In the previous post, we discussed indefinite integration. Now we shall discuss definite integration

► Definite Integration- We already know that   \int {f\left( x \right){\rm{ }}dx = g\left( x \right) + c}    \leftarrow  this c here is an integral constant. we are not sure about its value. This c is the reason we call this process indefinite integration. But suppose we do our integration between certain limits like:-

\int\limits_a^b {f(x)dx = \left[ {g(x) + c} \right]} _a^b   here a \to  lower limit while b \to  higher limit

\int\limits_a^b {f(x)dx = \left[ {g(b) + c} \right]} - \left[ {g(a) + c} \right]


You can clearly see that this function is independent of ‘c’. Means we can be sure about its value so this type of integration is called  Definite Integration.

►Definite Integration of a function f(x) is possible in [a,b] if f(x) is continuous in the given interval

►If f(x), the integrand, is not continuous for a given value of x then it doesn’t mean that g(x), the integral, is also discontinuous for that value of x.

► Definite integration of a function between given limits like     \int\limits_a^b {f\left( x \right)dx} \Rightarrow         Algebraic sum of areas bounded by the given curve f(x) and given lines x=a and x=b. That’s why the answer for definite integration problems is a single number.

► If \int\limits_a^b {f\left( x \right)dx} = 0 that shows a few things:-

(i) The lines between which area is bounded are co-incident(a=b)

(ii) Area covered above the x-axis=Area covered below the x-axis that means positive part of area and negative part of area is equal

(iii) there must be at least one solution/root to f(x) between x=a and x=b(this is something we study in ROLE’S THEOREM in detail)

► If given function f(x) is not continuous at x=c then we should write

\int\limits_a^b {f\left( x \right)dx} = \int\limits_a^{{c^ - }} {f(x)dx} + \int\limits_{{c^ + }}^a {f(x)dx}

► If given function f(x) > or <0 in any given interval (a,b) then  \int\limits_a^b {f\left( x \right)dx} \Rightarrow  >0 or <0 in given interval (a,b)

► If given function f(x)  \ge  g(x) in the given interval (a,b) then    \int\limits_a^b {f(x)dx \ge } \int\limits_a^b {g(x) \ge } dx 

in the given interval

► If we integrate the given function f(x) in the given interval (a,b) then

\int\limits_a^b {f(x)dx \le } \left| {\int\limits_a^b {g(x) \ge } dx} \right| \le \int\limits_a^b {\left| {f(x)} \right|dx}

<img src="definite integration.jpg" alt="definite integration">

Some More Properties of Definite Integration:- Read more

IB Mathematics Tutors- types of mathematical function(part-1)

IB Mathematics Tutors should give a fair number of hours in teaching functions.This is my third article on functions in the series of ib mathematics

For the sake of a comprehensive discussion, some standard functions and their graphs are discussed here

For the sake of a comprehensive discussion, some standard functions and their graphs are discussed here
1.Constant Function-
Domain= ℝ    

                                            Range    = {k}                                       

<img src="constant-function.jpg" alt="constant-function">


Range    = {k}

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IB Maths (Part-1)- Functions-An Introduction to functions in Mathematics

An Introduction to functions in Mathematics

In IB Maths both HL and SL, functions are one of the most important areas of mathematics because they lie at the heart of much of mathematical analysis. The concept of function is easy to understand.
Suppose I say that:


where x∈R. This says that y depends on x this can be said that y is a function of x. or


We can also say that x=√y here x depends on y or x is a function in y or


so it can be said that a function is an operator which takes an input and gives an output.The input is called independent variable while output is called dependent variable

<img src="functions.jpg" alt="functions">

ib elite tutor explained functions

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How to Get 7 in Ib Maths hl

How to Get 7 in Ib Maths hl

Many of my students got “7” in their IB Maths HL paper people so I believe my suggestion would be useful regarding this-

I think that students should solve their problems in one go. I don’t believe in following some “10 minute before your sleep” tips. All your hard work will go in vain if you forget chapter 3 while doing chapter 4 and because many times the hard questions a require knowledge across many chapters. What you need is a period of intense and thorough revision, in which your free study time is totally devoted to Math HL.

In the process of revision, Students should study the sections one after another.Textbooks split chapters into different sections with practice questions given at the end of each section.If you do not have such a textbook, please buy a good and I think there should be NO “I don’t know how to buy” issue).

Students should follow these instructions-

Instruction 1. Read the whole body of the section carefully.Students should understand it in the best possible way

Instruction 2.Try the practice questions in the order of their difficulty should not waste time on questions which you know, are very easy for you.Make a list of the questions you can’t do.Once the number of the listed questions reached 25-30% of the whole practice set, you should stop solving the questions and go back and study the body text because there is surely some concept that you are still missing.Once you are done with your reading, you can try your listed questions again.If the problem persists, you need expert should go to your teacher or seek online help at ib elite tutor. Read more

IB Diploma Programme,Middle year programme and primary year programme subjects

img src="img/subjects.jpg"

In primary year programme (PYP) an IB student can opt for following subjects:

1 Mathematics

2 Sciences and Technology
3 Art

4 Social Studies

5 Personal,Social and Physical Education

6 Languages

In middle year programme (MYP) an IB student can opt for following subjects:

  1.  Mathematics
  2. Sciences a)Physics              b)Chemistry
  3. Arts (performing and visual arts)
  4. Humanities a)History   b)Geography
  5. Physical Education
  6. Technology
  7. Language one
  8. Language two

Students who pursue the IB diploma must take six subjects:one each from Groups 1–5 and either one from Group 6 or a permitted substitute from one of the other groups, Three or four subjects must be taken at Higher level (HL) and the rest at Standard level (SL)

Here are the subject groups offered by IB
First Group:Studies in language and literature. Taken at either SL or HL, this is generally the student’s local language, with over 80 languages available. As of courses starting in summer 2011, there are two options for Language A, Literature, which is very similar to the old course, and Language and Literature, a slightly more difficult for.

Second Group:  An additional language, taen at the following levels: Language B (SL or HL), or Language ab initio (SL only).

Third Level: Individuals and societies. Humanities and social sciences courses offered at both SL and HL.Business and Management, Geography, Economics, History, I.T(information technology) in a Global Society, Psychology, philosophy and Social and Cultural Anthropology and World Politics.The course named as World Religions and the inter-disciplinary course Environmental Systems and Societies are also offered at SL level only.

Fourth Group: Chemistry, Biology, Physics, Design Technology and Computer Science are offered at both HL and Sl levels.The course Sport, Exercise and Health Science and the interdisciplinary course Environmental Systems and Societies are offered at SL only

Fifth Group: Mathematics. In order of increasing difficulty, the courses offered are Mathematics Studies, Mathematics SL and HL, and Further Mathematics SL and HL. The computer science courses were moved to Group 4 as a full course from first examinations in 2014

Sixth Group: The arts. Courses offered at both SL and HL: Dance, foreign languages, Music, Theatre, Visual Arts, and Film.Instead of taking a Group 6 course, students may choose to take an additional course from Groups 1–4 or either Further Mathematics HL (if already studying Mathematics HL) a course for Computer science from Group 4, or course based on school syllabus approved by IB.

Ib is also designing an online diploma course, I will discuss that in my next article.



Ib Internal assesssment

The Internal Assessment (IA) is unique to the International Baccalaureate. It is not a component of the Advanced Placement, the GCSE, or any other curriculum that I am aware of. The IA can be a paper, project, oral exam, workbook, or series of experiments, depending on the individual IB class. The IA is heavily criterion-referenced and is marked internally by the course instructor. The IB then selects a sample of all completed IA’s per class and these are forwarded to IB Examiners throughout the world for “moderation.” Moderation in this case means reviewing the accuracy of the internal marking. The examiner reviews each IA from the sample and then assigns his mark out of a total number of marks achievable – which varies from course to course. Even though the IA is marked internally by the instructor, it is the moderated mark by the examiner that is awarded to the student. Grades are then extended to all students completing the IA. For example, if a paper is rated internally at a 20 and is moderated at an 18, then all papers rated at a 20 (those not included in the sample) will receive a score of 18. Experienced IB teachers who have been through prior IA moderation are often very accurate criterion-referenced markers. In such cases, the moderator will simply confirm the teacher’s internal marking with an occasional point added or taken away. IB moderators must also deal with the internal marking of inexperienced teachers or those who pay little heed to IB IA criteria. In such cases, the moderator must read the entire sample closely and in effect, “re-mark” the student work. Poor internal marking generally results in inflated grades awarded by the teacher. It is an unfortunate fact the students are severely penalized in moderation when an entire sample is flawed by design or by inaccurate internal marking. The IA is a unique partnership between the student, the instructor, and the IB examiner. While neither the instructor nor the student ever has contact with the examiner, all are working from the exact set of criteria in constructing, guiding, and evaluating the finished product. One obvious characteristic of the IA is that the student, the instructor, and the school itself have clear control over much of the outcome of the IA. In other words, IA marks will generally be high when the instructor, the student and the available resources align to create a quality product. The more competent the instructor, the more motivated the student, the more subject-specific resources available, the higher the final mark will be. The outcome of IB exams (on the other hand) are controllable only to the extent that the material is adequately covered in breadth and depth and that students have highly developed test-taking skills. Students do not know the exact questions in advance when they sit for IB exams. However, students and instructors are well-aware of IA criteria, and the challenge is then to produce a quality product rather than to try and outguess the IB exam makers. As a consequence, students and instructors must strive to achieve the highest possible marks on the IA (which generally counts from 20% to 40% of the subject grade) in order to offset any problems that may be encountered on the IB exams. Top-tier public and private IB schools are generally consistent in the quality of their IA samples,  2 the high marks their students receive on their internally assessed work. IB Coordinators work hard to achieve internal consistency and high marks on all samples submitted. Top IA scores correlate with high exam marks and also with the Diploma pass-rate. Thus the IA is a solid indicator of the quality of the entire Diploma Program itself. Below the top tier schools, there is often a variation in the quality of IA samples from class to class. While the IBH Theater Arts mean IA score may be a 6 in any given year, the IBH Physics mean score may only be a 3. There are many reasons why IA scores may vary greatly between subjects at a particular school, not the least of which may be the nature of the subject itself. However, in my experience, a competent and knowledgeable IB teacher will consistently produce high IA scores year after year while an inexperienced teacher or one who is lacking in subject knowledge will struggle to produce passing IA scores. Many diplomas are lost each year by students who submit a sub-standard Internal Assessment in one subject or another. As a past IB teacher and Coordinator and a current IB Examiner, here are my 10 best tips on how to “ace” the Internal Assessment and help insure a top grade in any class:

1. All IA are criterion-referenced, meaning that the Examiner must insure that you satisfy a stated criterion before awarding top points. Criterion points are awarded according to markbands. A typical markband may be from 0-3 points with 3 indicating a fully-met set of criteria and 0 indicating an absence of any reference to the criteria. Know the IB criteria for your particular course – they are available in the IB Subject Guides – and continually review your work to insure that all criteria are being met.

2. Unless your instructor assigns your IA topic, choose your topic very carefully. Examiners often witness the same papers, experiments, and oral arguments year after year. Although the IB does not award points per se for creative and thoroughly original work, examiners will be duly impressed with an inquiry that is vibrant and fresh. Pursuing a boilerplate topic or simply rehashing an old subject does not constitute a meritorious inquiry. As a wise man once said, “you cannot fix by analysis what you ruined by design.”

3. Start your IA early, even before the instructor asks you to begin. Gathering your data or creating your portfolio is a time-consuming process. IA’s begun and finished in the senior year only can be hasty affairs. There are many sound reasons to begin your IA in your junior year and then do the bulk of your work during the summer between grades 11 and 12.

4. Personalize your IA wherever and whenever you can. Become a part of the inquiry itself. Write/present/create with clarity and enthusiasm. Make the examiner understand that the topic of your IA is your passion, regardless of what subject.

5. Consider the IA to be an open-ended rather than a closed inquiry. You don’t need to know how your IA will turn out or what you will conclude until your analysis is Pennsylvania International Academy | 3 complete. It is one thing to have a focus (essential for the IA) but it is another thing entirely to reach your overall conclusion before collecting your data.

6. Most IA’s whether written or oral ask you to develop your thesis or hypothesis before proceeding with your inquiry. Make certain this foundation statement allows for a legitimate quest rather than a dead-end pursuit. Intending to solve the Kennedy assassination once-and-for-all is doomed to failure. As would be to establish the precise impact of global warming on the world ecology. Don’t be afraid at the end of the IA quest to disprove your own theory or hypothesis. Some of the best IA’s uncover new information that disproves conventional wisdom or prior findings.

7. When you get stuck, ask for help. Sometimes a student gets so wrapped-up in the IA that they fail to see an answer that lays straight ahead. An objective secondopinion from a teacher or peer can keep you from working in isolation or going off on a tangent.

8. Bias is unavoidable but it must be controlled for throughout your inquiry. Personal opinions must also not be passed off as analysis. If your World Religion paper favors one religion over another, you will not earn many points from the examiner. If your Individual Oral Commentary in Humanities asks you to critique Hamlet and you choose to critique The Bard himself because you dislike Shakespeare, you will also lose major points. Empiricism is compromised if you exhibit your biases or prejudices. A personal or political agenda seeping into your work is a recipe for disaster.

9. Use IA templates. Chances are that your IB teacher has examples of winning IA’s from past years. Study these examples to learn what works and what doesn’t for the IA. Your teacher also has access to “Exemplars” of the Internal Assessment that he may download from the IB Online Curriculum Center. These typical examples run the gamut from stellar to mediocre. Most come with Moderator Comments so that you may see exactly where criterion points were won or lost.

10. Finally, the visual presentation that your IA makes is very important. Likewise important is the oral presentation you make on the IOC in Groups 1 and 2. Stylistically, put your best foot forward, whether you are submitting a paper, a portfolio, a project, a presentation or your Group 4 experiments. The closer your work is to publication quality, the more you improve your chances of receiving a high mark. Now go out there and grab that big “7” on the Internal Assessment!