Category Archives: Integration

Antidifferentiation

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Since a lot of classes start integration with antidifferentiation, I’ll discuss that first. If you decide to go with one of the other plans mentioned in my last post, then file this away for later.
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The key to finding antiderivatives is pattern recognition.

The simplest integrals are those that follow directly from derivatives such as. Students just need to recognize that the cosine is the derivative of the sine so \int{\cos \left( x \right)dx}=\sin \left( x \right)+C. We wish they were all that simple.

Something like \int{\tan \left( x \right){{\sec }^{2}}\left( x \right)dx} is more complicated. Is the integrand the derivative of \tfrac{1}{2}{{\tan }^{2}}\left( x \right)  or of \tfrac{1}{2}{{\sec }^{2}}\left( x \right)? The answer is yes.

The method called u-substitution helps in identifying the pattern of the integrand.  To use this method identify some part of the integrand as a function which you call u and then calculate du and hope that the du is the rest of the integrand.

For \int{\tan \left( x \right){{\sec }^{2}}\left( x \right)dx} you can try u = tan(x) and the du = sec2(x)dx and after making the substitutions:

\int{\tan \left( x \right){{\sec }^{2}}\left( x \right)dx}=\int{udu}=\tfrac{1}{2}{{u}^{2}}=\tfrac{1}{2}{{\tan }^{2}}\left( x \right)+C.

On the other hand we could try u = sec(x) so that du = tan(x)sec(x)dx and

\int{\tan \left( x \right){{\sec }^{2}}\left( x \right)dx}=\int{\sec (x)\left( \tan (x)\sec (x)dx \right)}

=\int{udu}=\tfrac{1}{2}{{u}^{2}}=\tfrac{1}{2}{{\sec }^{2}}\left( x \right)+C*

(The + C* here not the same as the + C in the expression above: C* = C – ½.)

Either way students need to recognize that part of the integrand is the chain rule contribution to the derivative of the other part of the integrand. The usual three steps to acquire these pattern recognition skills are practice, practice, practice.

Another detail of u-substitution is handling missing constants. For example, \int{\cos (2x)dx} .

Make the substitution u = 2x and then calculate du = 2dx so dx=\tfrac{1}{2}du. Then making these substitutions

\int{\cos (2x)dx}=\int{\cos \left( u \right)\left( \tfrac{1}{2}du \right)=}\tfrac{1}{2}\int{\cos \left( u \right)du=\tfrac{1}{2}\sin \left( u \right)}.

Then back-substituting: \int{\cos (2x)dx}=\tfrac{1}{2}\sin \left( 2x \right)+C.

For simple u-substitutions like this I find it easier to think, and not write, u = 2x so du = 2dx and then write this multiplying by ½ outside the integral sign to account for the extra factor of 2.

\int{\cos \left( 2x \right)dx=\tfrac{1}{2}\int{\cos \left( 2x \right)\left( 2dx \right)}}=\tfrac{1}{2}\sin \left( 2x \right)+C.

This idea only works with missing constants, since only constants can be moved in and out of integrands.

I explain the need for the constant to students by saying that the 2 appears (as if by magic) from the Chain Rule when you differentiate and therefore it must be present to disappear back into the Chain Rule when you antidifferentiate.

Point out to students that integration is very different than differentiation. Differentiation is pretty straightforward; you know what to do when you need to find a derivative. Antidifferentiation is more complicated since recognizing the form or pattern is necessary. The simpler looking integral \int{\ln \left( x \right)}dx is really more difficult than \int{\tfrac{\ln \left( x \right)}{x}dx}.

Finally, if you are teaching antiderivatives before beginning integration, when you get to definite integrals, you will have to remember to show students how to handle the limits of integration by using the same substitution.

For Advanced Placement AB calculus courses these (integrals that follow directly from derivatives and u-substitutions) are the only methods of integration tested. BC students also need to know integration by parts and partial fraction decomposition. I will discuss these later.

Integration Itinerary

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Before I write about integration, I’d like to say a few words about the order of topics. Since I assume most of you are AP Calculus teachers, the list of concepts and topics goes something like this:

  • Area of a region between the graph and the x-axis, which leads to
  • Riemann sums, which leads to
  • The definition of the definitive integral.
  • Numerical integration – left-, right-, upper-, lower-, midpoint sums. The Trapezoidal Rule and Trapezoidal approximations. Integration with technology.
  • The Fundamental Theorem of Calculus (FTC)
  • Applications of Integration
  • Introduction to differential equations.

Your preferred order of topics may be slightly different.

What is missing from this list is antidifferentiation or techniques of integration. There are several places where teachers have placed it with successful results.

      1. Many books and therefore many teachers place the topic of antidifferentiation at the very end of the derivative work or the very beginning of the integration. This has the advantage of having your students know how to do at least simple antidifferentiation when they need it right after the FTC. On the other hand, at this point students may not see the need for it and see it as the “game” of reverse differentiation.
      2. A more natural place to tackle it is immediately after the FTC. The advantage here is that now students have a reason to want to antidifferentiate so they can evaluate definite integrals.
      3. Other teachers change the order above slightly and teach applications before the FTC. They set up Riemann sums and definite integrals for the various applications and then have the student evaluate the integrals on their calculators.
        In the usual order listed above each application problem has two parts: the first is to set up the integral and then the antidifferentiation to evaluate it. Often the antidifferentiation is the longer part. By using technology to do the evaluation, students need only concentrate on setting up the correct definite integral and quickly do the evaluation on their calculator.
        Once the students are good at applications they then go on to the FTC and antidifferentiation as separate new topics. While learning antidifferentiation techniques teachers can assign one or two applications each night so students get more practice (spiraling).

I do not advocate one or the other of these approaches. They all have been tried and they all work. I am just pointing out the different ways so you will know that there is a choice. Pick the order you are comfortable with or pick a new order if you want to try something different.

Real “Real-life” Graph Reading

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A few days ago, Paul Krugman wrote a blog about the job situation in the US.   Evan J. Romer, a mathematics teacher from Conklin, NY, used it as the basis for a great exercise on reading the graph of the derivative, the subject of my last post. He posted the questions to the AP Calculus Learning Community.  I liked them so much I have included them on my blog with Evan’s kind permission. The questions and solutions are here and on the Resources Tab above.

He used the graph below which shows the change in the number of non-farm jobs per month; in other words, the graph of a derivative of the number of people employed. The jagged graph is the data; the smooth graph is a model approximating the data.

The model is \displaystyle C\left( t \right)=\frac{1000{{t}^{2}}}{{{t}^{2}}+40}-800

From this Mr. Romer developed a series of questions very similar to AP questions.  Don’t overlook the last note which discusses a “classic AP calculus mistake” made by the first person to reply to Krugman’s blog.

The first of Romer’s questions are integration questions, which you may not yet have gotten to with your class. Below is another graph of nearly the same data displayed as a bar graph. (Around February 2010 this was dubbed the “Bikini Graph” – if you look at the graph before that date you will see why.) It may be helpful in explaining the first of Romer’s questions to your class since each bar represents the change in the number of jobs for that month and leads into the concept of accumulation and the integral as the area between the graph and the axis. You can return to this when you introduce integration.

Thank You Evan.

Absolutely

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Absolute Value

The majority of students learn about absolute value long before high school. That is, they learn a lot of wrong things about absolute value.

  • They learn that “the absolute value of a number is the number without its sign” or some such nonsense. All numbers, except zero have a sign!  This sort of works with numbers, but becomes a problem when variables appear. True or false | x | = x? True or false | –x | = x? Most kids will say they are both true; in fact, as you know, they are both false.
  • They also learn that “the absolute value of a number is its distance from zero on the number line.” True and works for numbers, but what about variables?
  • They learn that “the absolute value of a number is the larger of the number and its opposite.” True again. How do you use it with variables?
  • They learn \left| x \right|=\sqrt{{{x}^{2}}} which is correct, useful for order-of-operation practice, and useful in other ways later, But they still compute \sqrt{{{\left( -3 \right)}^{2}}}=-3 and  \sqrt{{{x}^{2}}}=x since the square and square “cancel each other out.”

So here is a good vertical team topic. Get to those teachers in elementary and middle school and be sure they are not doing any of the above. They should start with the correct definition in words:

  • The absolute value of a negative number is its opposite.
  • The absolute value of a positive number (or zero) number is the same number.

This works all the time and will continue to work all the time. Teaching anything else will eventually require unlearning what they are using, and unlearning is far more difficult than learning.

When they start using variables and reading symbols translated into English, then the definition becomes their first piecewise define function:

  • \text{ If }x\ge 0,\text{ then }\ \left| x \right|=x;  and if x<0,\text{ then }\left| x \right|=-x
  • \left| x \right|=\left\{ \begin{matrix} x & \text{ if }x\ge 0 \\ -x & \text{ if }x<0 \\ \end{matrix} \right.

When reading this definition be sure to say “the opposite of the number” not “negative x” which in this case is probably a positive number.

Give variations of the two True-False questions above on every quiz and test until everyone gets it right!

When you see absolute value bars and want to be rid of them the first question to ask is, “Is the argument positive or negative? “Any time there is an absolute value situation, this is the way to proceed.

And yes, this does show up on the AP Calculus exams. Consider \int_{0}^{1}{\left| x-1 \right|dx} which appeared as a multiple-choice question a few years ago. Give it a try before reading on.

On the interval of integration, [0,1], \left( x-1 \right)\le 0 so \left| x-1 \right|=-\left( x-1 \right)

\displaystyle \int_{0}^{1}{\left| x-1 \right|dx}=\int_{0}^{1}{-\left( x-1 \right)}dx=\left. -\tfrac{1}{2}{{x}^{2}}+x \right|_{0}^{1}=-\tfrac{1}{2}+1-0=\tfrac{1}{2}

Now try \displaystyle \int_{0}^{1}{\sqrt{{{x}^{2}}-2x+1}\,dx}, or did we do this one already?

The First Week

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After reading my post on “The First and Second Day of School” Paul A. Foerster,  was nice enough to share  these problems from a recent presentation. They give a taste of derivatives and integrals in the first week of school and get the kids into calculus right off the bat.

The First Week of AP Calculus

Paul who recently retired after 50 (!) years of teaching, is Teacher Emeritus of Mathematics of Alamo High Heights School in San Antonio, Texas. He is the author of several textbooks including Calculus: Concepts and Applications, Second Edition, 2005, published by Kendall Hunt Publishing Company, www.kendallhunt.com (Formerly published by Key Curriculum Press).

Many years ago (almost 50) I remember teaching from his fine Trigonometry book . 

Thank you, Paul!

Why Limits?

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There are four important things before calculus and in beginning calculus for which we need the concept of limit.

    1. The first is continuity. Most of the time in pre-calculus mathematics and in the calculus we deal with nice continuous functions or functions that are not continuous at just a few points. Limits give us the vocabulary and the mathematics necessary to describe and deal with discontinuities of functions. Historically, the modern (delta-epsilon) definition of limit comes out of Weierstrass’ definition of continuity.
    2. Asymptotes: A vertical asymptote is the graphical feature of function at a point where its limit equals positive or negative infinity. A horizontal asymptote is the (finite) limit of a function as x approaches positive or negative infinity.

Ideally, one would hope that students have seen these phenomena and have used the terms limit and continuity informally before they study calculus. This is where the study of calculus starts. The next two items are studied in calculus and are based heavily on limit.

3. The tangent line problem. The definition of the derivative as the limit of the slope of a secant line to a graph is the first of the two basic ideas of the calculus. This single idea is the basis for all the concepts and applications of differential calculus.

4. The area problem. Using limits it is possible to find the area of a region with a curved side, even if the curve is not something simple like a semi-circle. The definite integral is defined as the limit of a Riemann sum and gives the area regions with a curved side. This then can be extended to huge number of very practical applications many having nothing to do with area.

So these are the main ways that limits are used in beginning calculus. Students need a good visual  understanding, what the graph looks like,  of the first two situations listed above and how limits describe and define them. This is also necessary later when third and fourth come up.