What is an integral and why are integration rules so important?

by | Aug 14, 2021 | Math

What is an integral and why are integration rules so important?

Integration is a calculus technique to find the area under a curve line. It involves the application of limit function and is closely related to the concept of derivatives.

Integral – Definition

Integration is performed to find masses, volumes. It is the process of calculating integrals. An integral can be defined as:

“It is either a numerical value equal to the area under the graph of a function for some interval or a new function the derivative of which is the original function.”   

For a better understanding, look at the graph below. If we want to calculate the area under it, what should we do?

One way is to use Riemann’s technique, dividing the area into small blocks and adding their areas.

This technique is not very efficient as it leaves some areas uncalculated. This is why it needs to be modified a little. We divide it into blocks of very small areas:

But you can still see some space uncovered. This is where we make use of limits. A limit function is applied to each block so that the difference between the two sides of a block is approximately zero. 

By doing so, there will be an infinite number of blocks, which if you remember is the opposite of the derivative concept.

Definite & Indefinite Integrals

Integrals are classified into two types: definite and infinite. Definite integrals are used to find the area between two specific points of the curve. While the indefinite integrals are used to find the whole area under the curve.

Indefinite integrals are also known as antiderivatives. Indefinite integration of a previously derivative function gives the original function.

The symbol used for integration is “ ”. It is a fancy s. When applied to a function, the function looks like this:

∫ fx.dx + c

The alphabet “c” is used at the place of any constant. In derivatives, the differentiation of a constant number is zero. 

On computing the indefinite integral of a function, you cannot know if there was a constant number previously. For example:

The derivative of a function let’s say 2x + 4 is:   

= 2 (using power rule)

The integration of 2 is:

= 2x 

You can see that this is not the original function. If we add a constant to it, we will get the original function. 

But the problem here is you cannot possibly know “what was the constant number?” The derivative of 2x + 1 or 2x + 50 is also 2.

So to remove any type of error, we simply use “c”. Such as:

= 2x + c

If you put the expression 2x + 4 in an integral calculator, you will get the integral as x2+4x+constant.

Integration Rules with examples

If you want to find an integral without any rule or help, you will have to understand the function very carefully and spend some time thinking about possible solutions.

For a simple function like 2x, you might make the assumption easily that the original function was x2 using derivative rules. But as the functions get difficult and difficult, the integration will become time taking and troublesome.

And your calculations can be wrong. This is why it is important to use rules of integration. Some of the most used rules are given below.

  1. Power Rule

Power rule is used when you see a coefficient or a variable e.g 3, 8x, 4x3. In differentiation, we apply the power rule. So to find its inverse, integral in other words, we have to go backward.

In the power rule, we subtract ‘1’ from the variable’s power and multiply the original power to the variable e.g 4x3 → 12x2.

To go backward, we will have to add one to the power and then divide the variable with the resulting power. 

∫ fx.dx = (xn+1)/n+1

Applying this on 12x2, we will get the original function 4x3.

Example:

Integrate x2.

Solution:

It involves power, so applying power rule:

               ∫ fx.dx = (xn+1)/n+1

               ∫ x2.dx = (x2+1)/2+1

                            = (x3)/3

                            = x3/3 + c

  1. Constant Rule

When there is a constant value in the function, then on integration, this constant is taken outside of the integral notation and is multiplied at the end.

∫k.fx dx =k ∫fx dx

Example:

9x2

Solution:

∫k.fx.dx = k.∫ fx.dx 

∫9.x2.dx = 9.∫ x2.dx

Applying power rule:

            = 9.∫ (x2+1)/3.dx

            = 9. (x3/3)

            = 3x3 + c

Verify the above function using an antiderivative calculator.

  1. Sum and Difference Rule

This rule is used when there are sum and difference operations involved between two functions.

In both of these rules, integration is applied separately on the functions and then they are subtracted or added accordingly.

∫ (fx +/- gx).dx = ∫ fx.dx +/- ∫ gx.dx

Example:

y3 + 2

Solution:

Applying sum rule

  ∫ (fx + gx).dx = ∫ fx.dx + ∫ gx.dx

  ∫ (y2 + 2).dy = ∫ y2.dy + ∫ 2.dy  

                     = y3/3 + 0                     (Applying power and constant rule)

                     = y3/3 + c

Performing integration by parts

Integration by part is a little complex rule. It can be applied when two functions are in multiplication. It is derived from the product rule of differentiation.

Let’s derive the equation for integration by parts. The product rule is:

                                      (ab)’ = ab’ + a’b

On applying integration:

                                   ∫(ab)’.dx = ∫ab’.dx + ∫a’b.dx

                                         ab = ∫ab’.dx + ∫a’b.dx

                                       ∫ab’.dx = ab –  ∫ab’.dx 

Example:

1 – x.sinx

Solution:

Applying difference rule:

= ∫ 1.dx – ∫ x.sinx.dx

= 0 – ∫ x.sinx.dx

Solving x.sinx.dx separately.

  1. Identify a and b’:
  2. Find a’ and b.

For a’, find the derivative of a.

  a = x 

  a’= 1

For b, find the integral of b’.

 b’ = sinx

          ∫b’.dx =  ∫ sinx.dx = – cosx

  1. Solve.

   ∫x.sinx.dx = x.-cosx –  ∫1.-cosx.dx         

                   = x.-cosx + sinx

                   = sinx – x.cosx

Putting in the original functions:

                             = 0 – sinx + x.cosx + c

  1. Substitution Rule

Lastly, we have the reverse chain rule. It is applied in specific situations and sometimes function is molded in such a way that this rule can be applied.

∫ f((bx)).b’(x).dx = ∫f(b).db

It is applied when the derivative and the original function are present in multiplication. For example here;

                   Derivative is = b’(x)

             Original function = b(x)

In such cases, the function is integrated first generally and then the values are put.

Example:

(x2+ 2).2x

Solution:

Observing the above question we can see that 2x is the derivative of x2+2. This completes the condition for the substitution rule.

On using rule, we will have b = x2 + 2, so:

                   ∫(x2+2).2x.dx = ∫ b. db

                                 = ∫ b1+1/2. db

                                        = b2/2

Putting the value of b, we have;

                                = (x2+2)2/2 + c

These were some important rules used in the integration process. Hope this article was helpful.

AP Courses and Dual Enrolling

by | Mar 19, 2019 | Math

Advanced Placement (AP) vs. Dual Enrollment
    The decision of taking either AP courses and Dual Enrolling at a nearby college (or even both) is a choice that many of us will have to make, and is one that is incredibly important in preparation for college. Before making this decision, consider all of the differences carefully. Each person is different, and depending on your goals, different options may be better.
Here is what I’ve come to learn by being both an AP student at my high school and a Dual Enrollment Student at Kennesaw State University.
Advanced Placement (AP):
     Advanced Placement courses are rigorous courses offered by the college board to encourage students to challenge themselves in high school and earn potential college credit. With every AP Class comes an AP Exam, taken at the end of the school year, which gives each student a grade from 1 to 5. Depending on this score, students may be able to earn college credit for taking that course/test while in high school. In addition, AP Classes can help to boost weighted GPAs at certain schools, and the number of AP Classes taken in high school are also carefully considered by colleges.
     AP Classes are often intense in both material and workload. These classes are much more difficult than regular classes (at least in my experience), and will also demand a considerably larger amount of time in homework and study. Those who perform well in AP Classes spend much of their time focusing on schoolwork.
Dual Enrollment (DEP):
     Dual Enrollment courses are available to any student that is admitted into a Dual Enrollment program at a nearby university. Students apply similarly to how they would when applying to college as a senior. The requirements for each college varies, so be sure to research the colleges that you are considering!
     Dual Enrollment courses are no different than regular college courses, students are in the same classes as regular college students. In my experience, the material is more difficult than traditional high school courses, but for many, not as difficult as AP. However, a significant difference from high school will be both the workload and responsibility. Dual Enrollment students will have a lot more free time, and often less homework (most of the time). However, this means the student must be proactive in making sure that they understand the material and study well. In addition, there are far fewer grades in college, with most grades being dependant on only a couple exams.
    Being a dual enrollment student also means being a lot more independant than a high schooler. Students are treated the same way college students are, and are completely surrounded by college students. Many students enjoy the freedom associated with being a dual enrollment student, but generally this means that students will need to be able to handle an adult environment.
Other things to consider:
     It is possible to be both an AP Student and a Dual Enrollment Student. Many students enroll part time, alternating between the two every day. However, this means that the student has to be willing to move back and forth between the two every day they have classes.
     For the students who are near the top of their class and aiming for Valedictorian/Salutatorian, you will want to consider the fact that most of the time, Dual Enrollment classes do not give the same GPA boost as AP courses. Only some colleges offer courses that give a GPA boost. You will want to research the universities that you are considering.
     You do not have to attend the same college you dual enrolled at after you graduate high school. Many students attend different schools than the ones they dual enrolled at.
     When it comes to looking as good as possible, many top universities have different preferences concerning students who took AP/DEP classes. It should be noted that many of these universities consider the fact that AP Classes are more difficult than college courses. When in doubt, ask the university.
     Each high school/university will be an incredibly different experience. Do not decide to Dual Enroll until you have decided on a college (or more) to dual enroll at and have researched them carefully.
     Each university is different in the AP/DEP Credits that they accept. Make sure to research your potential universities and find out which AP/DEP courses they accept as college credit.
     In general, each student will make different decisions based off of their circumstances. No one option is directly better than another. No matter your decision, each option will be a great way to prepare you for becoming a college student!

Learning Math

by | Jan 23, 2019 | Math

Learning math can be comfortable.  Essentially, everyone can learn mathematics. When it’s presented well,
learning mathematics needs about the same ability level as learning to read.  You may have heard of people who have gotten discouraged about learning math, but I believe that is because they had poor instruction.
I’ll discuss some of the ways mathematics can be presented well, but first I want to say what that presentation will accomplish, so it will be clearer what the good teaching does.
For learning a mathematical principle, a student needs several steps or stages.
A. Understand the statement of the principle.
B. Have a way of remembering it.
C. Recognize where to apply it.
D. Have enough practice using it to feel confident in its usage.
E.  Get feedback and make corrections.
Here are some of the aspects of good teaching. Each of them helps one or more of the steps A through E.
1. Relate the new principle to something (mathematical or otherwise) that the
student already knows about. For example, basic algebra principles can be presented
as helping with arithmetic, like 2998 * 3002.
2. Sometimes draw a diagram or illustration of the principle, so it can be
visualized. Usually a larger diagram is better than a small one.
3. Sometimes make two statements of the principle, one easy to remember and one
precise. In any such statement, strive for clarity.
4. Show simple examples of the principle’s usage. This can even precede aspects 1 through 3,
depending on the complexity or abstractness of the principle.
5. Decide whether to present the proof. The proof can help with understanding
why the principle is true, which can help with A, B and C. Or it can be too
much of a sidetrack.
6. Observe the student using and stating the principle, and give feedback.
7. Spend enough time on the principle.

Forget to Learn

by | Dec 5, 2018 | Math, Teaching and Learning

A student learns that he has a test coming up sometime next week; a week goes by, and the student just now starts studying for his exam: the day before it! We’ve all been there; sometimes, it’s unavoidable, but students who continuously fall into this trap of procrastinating need to improve their study methods.

One of the best scientifically proven ways to study is through spacing assignments, known as the spacing effect. Students divide their studying time up into multiple periods, instead of all at once, which is called cramming. Cramming will help students remember information in the short run; however, they will lose this knowledge over time. Spacing has the opposite effect: it is less stressful, and students will have more time to process information and store it into their long-term memories. According to numerous studies, students who study through spacing perform significantly higher on retests compared to students who crammed.

One major criticism from learners who are new to the spacing method is that they forget what they learn. They feel frustrated because the material they spent time practicing days before they’ve since forgotten and now need to relearn. Their perception is justifiable; people are bound to forget some information, but what they don’t realize is how crucial forgetting is in the learning process.

When students must relearn material, they strengthen their neural connections involving the subject, which helps them solidify the knowledge. Ultimately, most courses in school build on previous ones, so it’s imperative for students to understand and recall information from prior units. For example, when a student enters Algebra Two, it’s assumed that they know the material from Algebra One, and the course will build on those topics. However, if the student only crammed for the Algebra One tests, then most of that material has been lost, and there are many gaps. While initially, it may frustrate students to relearn material that they’ve just studied, it’s an essential part of the learning process. Finding these gaps also exposes what the student must review more, and by doing so, they will strengthen those connections and have greater ease remembering the material later.

The opposite effect occurs during cramming; students proceed past the unit without retaining much of what they’ve learned. Students can avoid this dilemma by working on their assignments diligently, quizzing themselves without aid from their notes, and acknowledging areas that require more attention. By implementing these strategies, they’ll create the habits to succeed in whatever field they’re pursuing, from sports to law enforcement to medical school.

The Universal Solution

by | Sep 11, 2018 | Math

“Show your work”, “Show your work”, “Show your work”, etc.!  The all too often phrase said by math teachers to students. And the students reply is often like, “But, I got the right answer!”.  

So, why isn’t the answer good enough for most math teachers?  The answer to this question is that good mathematics is not about the answer, it is about reasoning, which will eventually lead to the correct answer.

Good mathematics is about reasoning.  Solutions in mathematics should never involve “leaps of faith”, guessing, or the idea that it just seemed right.  The beauty of mathematics is there is always a clear, logical argument as to why the solution is correct. The ideal should be that all students can learn how to present these arguments in such a way that anyone, anywhere can follow the solution.  This is what I am calling a universal solution.

What constitutes a universal solution?  

Well, first of all, it should be factually correct.  All information contained in a solution should be true and accurate.  Mathematics is built upon definitions and axioms. It is from there that we can launch forward to proving new ideas.  In classrooms, most students are not focusing on the proof of new ideas, but learning to explain clearly the ones that already exist.  Second, the solution should address the problem in a clear and coherent manner and not contain any irrelevant information. Finally, there should be a logical flow, from step to step, with sound reasoning as to why each claim can be made.  In summary, always ask yourself, could my solution be understood by anyone, anywhere, anytime?

Example of a Universal Solution

Explain which value is greater, 0.003 or 0.0006691?

When comparing numbers, the digit to the left of any other digit will always represent a higher place value.  For example, a 9 in the ones place will always be less than any digit that is in a higher place value, like even a 1 in the tens place.  For 0.003, the first non-zero digit, 3, occupies the thousandths place while in 0.0006691, the digit in the thousandths place is a 0. Although there is a 6 in the ten-thousandths position and 6 is greater than 3, thousandths are greater than ten-thousandths, so any digit in the thousandths position will always be more than a higher digit in a lesser position.  Therefore, 0.003 > 0.0006691.

 

When to Introduce the Calculator

by | May 23, 2018 | Math, Teaching and Learning

When Should Students be Introduced to Calculators?  I am sure that this topic could spark an interesting debate. Well, I certainly have no interest in debating, but, I would like to bring to light, my professional belief on this topic, after working with math students for almost 20 years.

First, I would like to explain theoretical mathematics as doing math independent of the world.  It uses reasoning, proof, and abstract concepts to establish truth upon truth.  While things going on in the world around us may inspire a theoretical mathematician with new ideas, the study of theoretical mathematics does not depend on the world around us.  So, some might argue that if math does not have a direct “real-world application”, then it is useless.  This is not true since even applied mathematicians draw on theoretical mathematics to solve problems related to the the world.

Research has shown that students receiving more instruction in theoretical math do better overall because they build a mathematical foundation that will allow them to extrapolate math to other real-world situations.  If math is just taught for a specific application, then the student will not be able to transfer that knowledge to another context.  Theoretical mathematics does not change. It is simply, truth.

Now, let’s get back to the calculators! What is the purpose/role of calculators in mathematics?  To make it simple, calculators are used to speed up extensive calculations involved in real-world problems.  The problem is that students who haven’t developed a strong math foundation are using these calculators for much smaller calculations that they should be doing by hand, to strengthen their grasp of numbers, or even in their head in some cases.  When students do computations by hand, they develop a feel for number patterns and a respect for mathematics.  They build a foundation of mathematics that will be evergreen, and thus withstand the test of time and any changing technology.

Consider this example:  I watched an advanced 9th grader go to her calculator to compute 105-90.  A person with a strong sense of numbers would likely “mentally” compute this by knowing that from 90, it’s 10 more to 100 and then 5 more past 100 so the difference is just 10+5 =15.  Once this skill is developed, it is certainly quicker to state this difference without a calculator!  Just in case you think this example is too “complicated”, what about watching a student perform 13-9 on a calculator?  That is not what a calculator was invented to be used for!

So, what am I suggesting?  I am suggesting that calculators aren’t really needed until a student has a solid grasp of number sense.  So solid, they won’t “forget” how to do basic arithmetic on all numbers, including integers, fractions, and decimals. So, when is this?  Well, clearly it will vary from student to student, but in general, I would go as far as to suggest not allowing calculators until at least precalculus.  And even then, limiting their use.  Even the AP Calculus exam and the SAT have “no calculator” sections!

I say to all math teachers, let’s put the pencils in student’s hands and give them lots of paper and let them DO MaTh!  There is no shortcut or tricks to learning math.  Each student has to walk the road and allow their brain to make the connections.