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Integration and Finding Areas

An image of a curve on graph paper with a shaded area under it and an integral formula for the areaIn order to calculate the area under a curve, we integrate its function between the boundaries [a,b] as shown in the image at left.  This means that it is vital that we develop our integration skills in order to become good at finding areas.

Once you have mastered the skill of integrating, there are a few other refinements to learn.  Here are some of them:

A.  How to find areas under curves (but above the x-axis).

B.  How to find areas between curves and the x-axis.

C.  How to find areas between curves.

D.  How to find areas under odd and even functions.

All will be explained as I add more material here.

First Things First ~ The Fundamental Theorem of Calculus!

If you are currently studying calculus at school or, if you never understood it but would like to see if it makes sense now, this video is designed for you. It is not designed for the complete novice. I assume some understanding of graphing and differentiation.  At a later time I intend to explain the foundations of calculus 'from the ground up.'

For many years (at least since Archimedes, around 250 BC), mathematicians had known that areas of unusual shapes could be calculated by imagining that they were divided into innumerable infinitesimally thin strips and adding all those 'parts' together. They had also known how to calculate gradients and rates (comparing two quantities by dividing them) and, with the development of calculus, how to calculate them anywhere on a curve by differentiating the function.

The Fundamental Theorem of Calculus showed that these two concepts were intimately related! In fact, finding the area under a curve proved to be the exact opposite to finding its gradient! That is, if the process for finding a gradient is called differentiating, the process for finding the area under a curve is called antidifferentiating.

Suddenly, this made finding areas much easier! Antidifferentiating is also called finding primitives or integrating.

This understanding is the result of the first part of the Fundamental Theorem of Calculus. The second part of the theorem shows that the exact area under a curve and between two boundary values on the x-axis is found by substituting those boundary values into the primitive and finding the difference between the two results.

Integrating Powers of X (x^n)

The Fundamental Theorem of Calculus shows us that integrating (finding areas under curves) is the inverse of differentiating. By examining how we differentiate powers of x, I show how this leads to TWO basic methods of integration for those functions! Although most people will prefer the second method for integrating powers of x (as I do), it is very helpful to learn and understand both methods … because the first method is more useful for solving more complicated integrals.

Therefore, I not only show the theory behind both methods, but demonstrate how to use them with a variety of functions. Please learn to construct and use both methods. You will benefit immensely by doing so.

Integrating Powers of a Function of x

By starting with the derivative of f(x)ⁿ we discover two methods for integrating powers of functions of x.

I explain how to derive the pattern for both forms ... so that you can learn those patterns, or rapidly derive them for youself whenever you need during exams. I then demonstate the use of both methods using two or three integrals each time.

Integrating Exponentials of a Function of x

There is really only one pattern for integrating expressions of this kind.

By starting with the derivative of e^f(x) we discover that the pattern we look for in an integral is e^f(x) preceded by the derivative of f(x) ... i.e. f'(x).e^f(x). Once we have that form (which may involve manipulating constants), the integral is amazingly easy to find!

I show you how to derive (or remind yourself of) the pattern during an exam, what to look for in an integral, and how to perform the integration using two examples.

Integrating Fractions that Produce Logarithmic Functions

This kind of integral looks daunting because it is always a fraction, and many students do not like fractions. Its structure, however is normally quite simple ... a function as the denominator and its derivative as the numerator. Just think of the fraction bar (vinculum) as separating the two!

As with previous videos, I begin with the derivative ... this time, of ln[f(x)] ... in order to discover the pattern. I then perform a couple of integrations to demonstrate how simple the use of this pattern really is.

It is true that there are complicated and difficult-to-analyse integrals of this kind. I have not bothered with them here. The most common difficulty arises when the function or its derivative produces fractions (or negative indices). In cases such as these I would certainly recommend using the substitution method in order to simplify and resolve the integral. For relatively straight-forward cases, however, I believe that knowing and using this pattern is quite sufficient.

Integrating Trigonometric Ratios of Functions of x

In this video, I deal with the three basic trigonometric functions together because the pattern is the same for all three. For simple integrations, there is no need to resort to using the substitution method. Using a simple pattern will be quite sufficient, provided you understand, recognise, and can use the pattern.  In other words, its use requires practice. The substitution method is an excellent tool but is of much more value for more difficult integrals.

I begin by 'deriving' the form of three integrals by using the chain rule ... for sin[f(x)], cos[f(x)], and tan[f(x)]. By reversing the process (that is, integrating, or antidifferentiating) I help you compare the patterns for each and learn 'what to look for' in an integral.

I then integrate three trigonometric expressions, explaining the process for each.

Integrating Powers of Trigonometric Ratios of x

This integral type looks daunting because students seeing large powers of sin(x), for example, can feel that the integral must, by necessity, be complicated. What I try to show is that if the integral conforms to a certain pattern, the solution is quite simple. Of course ... if it does not conform to the pattern, then the integral may be just as difficult as students suspect, or even worse!

The key is that if an integral contains a power of sin(x), cos(x) or tan(x), it must be multiplied by the derivative of that same function and nothing else (except for constants, since we can adjust or manipulate those). So, the pattern that we look for is of this form:

  • sec²(x).tanⁿ(x)
  • cos(x).sinⁿ(x)
  • -sin(x).cosⁿ(x)

In this video, I explain why these patterns must be as they are, how to reconstruct them during exams (if you have forgotten them), how to identify them in integrals, how to manipulate the constants, and demonstrate all this by integrating three expressions.

How to Integrate With Speed! ~ Mastering the Six Basic Patterns

If you have difficulty with integrating functions (calculus), I hope you will find that my three videos will help you.

  • In this first video I explain the six basic patterns that you need to master. I also explain why you may have been confused about two of the patterns, and then demonstrate how to find 'simple' integrals using a few examples.
  • In the second video I will show you how to integrate using the inverse of the Chain Rule. I will demonstrate a particular way of integrating that may appeal to you and make the process less confusing.
  • In the third video I will explain how to integrate by parts. This is the inverse of the Product Rule that we learn and use when differentiating. I will demonstrate the use of this pattern using a number of examples.

In none of these videos do I use substitution. I was taught these methods using substitution, and it is important that you know the process of substitution but ... when there is a recognisable pattern, I find that the substitution method 'bogs' students down in unnecessary detail and they get lost or make mistakes.

My method of teaching involves helping students understand patterns rather than rules. This is not to say that the rules aren't important. We need these to be convinced of the truth/validity of a process, and to understand it thoroughly. It is the pattern, however, that allows us to recognise and use a technique with ease and with a smaller risk of making careless mistakes. It is also much faster!

My experience is that many students get 'bogged down' in detail and lose sight of the overall strategy that they are following (or the overall structure of the problem). By teaching students to recognise and follow patterns, I find that their confidence, speed and accuracy (and understanding) all improve significantly. I hope these videos will help you in the same way.

Demonstration Video ~ 5 Speedy Integrals Using the Basic Patterns

Do you want to complete most integrals within 30 seconds or so?  (Note that this is not a guarantee ... simply a guideline.)

This video may help you do just that! It may not make sense to you, however, unless you have seen the above video.

The integrals are (with their starting times):

  • 00:32 ~ ∫(6x²+5√x³).dx
  • 02:04 ~ ∫(5e^x/3).dx
  • 02:37 ~ ∫2/(3x).dx
  • 03:02 ~ ∫3sinx.dx
  • 03:40 ~ ∫(sec²x/2).dx

How to Integrate With Speed ~ Recognise and Use the Chain Rule Pattern

If you have difficulty with integrating functions (calculus), I hope you will find that my three videos will help you.

  • In the first video I explained the six basic patterns that you need to master. I also explain why you may have been confused about two of the patterns, and then demonstrate how to find 'simple' integrals using a few examples.
  • In this second video I will show you how to integrate using the inverse of the Chain Rule. I will demonstrate a particular way of integrating that may appeal to you and make the process less confusing.
  • In the third video I will explain how to integrate by parts. This is the inverse of the Product Rule that we learn and use when differentiating. I will demonstrate the use of this pattern using a number of examples.

In none of these videos do I use substitution. I was taught these methods using substitution, and it is important that you know the process of substitution but ... when there is a recognisable pattern, I find that the substitution method 'bogs' students down in unnecessary detail and they get lost or make mistakes.

Demonstration Video ~ 11 Speedy Integrals Using the Chain Rule Pattern

In this video I integrate eleven functions using the the method and principles based on reversing the Chain Rule (see the video directly above this one).  The integrals are (with their starting times):

  • 00:20 ~ ∫(3x+4)^5.dx ~ ∫(3x-1)^(-2).dx ~ ∫³√(4x+3).dx
  • 04:19 ~ ∫x.sin(x²+1).dx ~ ∫sec²(6x).dx ~ ∫sinx.e^(cosx).dx
  • 06:43 ~ ∫2e^(3x+4).dx
  • 07:55 ~ ∫x²/(x³+2).dx
  • 08:32 ~ ∫(x-1)/(x²-2x).dx
  • 09:41 ~ ∫sec²x.(tanx)^5.dx
  • 10:52 ~ ∫sin²³x.cosx.dx

How to Integrate With Speed ~ Recognise and Use 'Integration by Parts' Pattern

If you have difficulty with integrating functions (calculus), I hope you will find that my three videos will help you.
This is the third one.

  • In the previous two videos I explained the six basic patterns that you need to master and then the six Chain Rule patterns that are based upon them. I also explained why you may have been confused about two of the pattern types, and demonstrated how to calculate a variety of integrals using these very recognisable patterns.
  • In this video I explain Integration by Parts.  This is the inverse of the Product Rule that we learn and use when differentiating. I also demonstrate the use of this pattern using a number of examples.

Demonstration Video ~ 6 Integrals Using the Integration by Parts Pattern

Integration by Parts is a less rapid integration process. Yet, even with all that this entails, you should be able to complete each basic integration within a minute. Even the more difficult ones should not take you more than a couple of minutes (unless you have to use Integration by Parts a third or fourth time).

This video may help you reach those goals/benchmarks! It may not make sense to you, however, unless you have seen the explanatory video directly above this one.  The integrals are (with their starting times):

  • 00:50 ~ ∫x.sec²x.dx
  • 02:34 ~ ∫x.lnx.dx
  • 06:08 ~ ∫sin(lnx).dx
  • 11:23 ~ ∫x²e^(4x).dx
  • 16:49 ~ ∫(cosx/sin²x).dx
  • 21:24 ~ ∫x².cosx.dx

You are an excellent teacher … Would have been a different world for me had I had one like you … (born 49) left school in 9th grd. Still love math and don’t understand people who do not … You lay knowledge on a silver platter and it is much appreciated …

gregsphere (on a CCM YouTube video about How to Subtract [Large] Numbers Easily)

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