# Type 5: Table and Riemann Sum Questions

Before we look at the table and Riemann sum problem take a look at this: A Google Employee has calculated $\displaystyle \pi$ to over a trillion decimal places, To be exact 31,425925,535,897 places…Hum, where have I seen those digits before?  And it only took took 25 virtual machines 121 days to do it.

On to Riemann sums and table problems:

Tables may be used to test a variety of ideas in calculus including analysis of functions, accumulation, theory and theorems, and position-velocity-acceleration, among others. Numbers and working with numbers is part of the Rule of Four and table problems are one way this is tested. This question often includes an equation in a  latter part of the problem that refers to the same situation.

What students should be able to do:

• Find the average rate of change over an interval
• Approximate the derivative using a difference quotient. Use the two values closest to the number at which you are approximating.  This amounts to finding the slope or rate of change. Show the quotient even if you can do the arithmetic in your head.
• Use Riemann sums (left, right, midpoint), or a trapezoidal approximation to approximate the value of a definite integral using values in the table (typically with uneven subintervals). The Trapezoidal Rule, per se, is not required; it is expected that students will add the areas of a small number of trapezoids without reference to a formula.
• Average value, average rate of change, Rolle’s theorem, the Mean Value Theorem and the Intermediate Value Theorem. (See 2007 AB 3 – four simple parts that could be multiple-choice questions; the mean on this question was 0.96 out of a possible 9.)
• These questions are usually presented in some context and answers should be in that context.
• Unit analysis.

Do’s and Don’ts

Do: Remember that you do not know what happens between the values in the table unless some other information is given. For example, do not assume that the largest number in the table is the maximum value of the function, or that the function is decreasing just because a value is less than the preceding value.

Do: Show what you are doing even if you can do it in your head. If you’re finding a slope, show the quotient.

Do Not do arithmetic: A long expression consisting entirely of numbers such as you get when doing a Riemann sum, does not need to be simplified in any way. If you a simplify correct answer incorrectly, you will lose credit.

Do Not leave expression such as R(3) – pull its numerical value from the table.

Do Not: Find a regression equation and then use that to answer parts of the question. While regression is perfectly good mathematics, regression equations are not one of the four things students may do with their calculator. Regression gives only an approximation of our function. The exam is testing whether students can work with numbers.

Free-response examples:

• 2007 AB 3 (4 simple parts on various theorems, yet the mean score was 0.96 out of 9),
• 2017 AB 1/BC 1, and AB 6,
• 2016 AB 1/BC 1

Multiple-choice questions from non-secure exams:

• 2012 AB 8, 86, 91
• 2012 BC 8, 81, 86  (81 and 86 are the same on both the AB and BC exams)

Schedule of review postings:

# Integration

### Integration –DON’T PANIC

As I’ve mentioned before, I try to stay a few weeks ahead of where I figure you are in the curriculum. So here. early in November, I start with integration. You probably don’t start integration until after Thanksgiving in early December. That’s about the midpoint of the year. Don’t wait too much longer. True, your kids are not differentiation experts (yet); there will be plenty of differentiation work while your teaching and learning integration. Spending too much time on differentiation will give you less time for integration and there is as much integration on the test as differentiation.

The first thing to decide is when to teach antidifferentiation (finding the function whose derivative you are given). Many books do this at the end of the last differentiation chapter or the first thing in the first integration chapter. Some teachers, myself included, prefer to wait until after presenting the Fundamental Theorem of Calculus (FTC). Still others wait until after teaching all the applications. The reasons  for this are discussed in more detail in the first post below, Integration Itinerary.

Integration itinerary – a discussion of when to teach antidifferentiation.

The following posts are on different antidifferentiation techniques.

Antidifferentiation u-substitution

Why Muss with the “+C”?

Good Question 12 – Parts with a Constant?

Arbitrary Ranges  Integrating inverse trigonometric functions.

Integration by Parts I (BC only)

Good Question 12 – Parts with a Constant  How come you don’t need the “+C”?

The next three posts discuss the tabular method in more detail. This is used when integration by parts must be used more than once. If memory serves, using integration by parts twice on the same function has never shown up on the AP exams. Just sayin’.

Integration by Parts II (BC only) The Tabular method.

Parts and More Parts   (BC only) More on the tabular method and on reduction formulas

Modified Tabular Integration  (BC only) With this you don’t need to make a table; it’s quicker than the tabular method and just as easy.

Revised and updated November 4, 2018

# Table & Riemann Sum Questions (Type 5)

Tables may be used to test a variety of ideas in calculus including analysis of functions, accumulation, position-velocity-acceleration, theory and theorems among others. Numbers and working with numbers is part of the Rule of Four and table problems are one way this is tested.

What students should be able to do:

• Find the average rate of change over an interval
• Approximate the derivative using a difference quotient. Use the two values closest to the number at which you are approximating.  This amounts to finding the slope. Show the quotient even if you can do the arithmetic in your head.
• Use Riemann sums (left, right, midpoint), or a trapezoidal approximation to approximate the value of a definite integral using values in the table (typically with uneven subintervals). The Trapezoidal Rule, per se, is not required; it is expected that students will add the areas of a small number of trapezoids without reference to a formula.
• Average value, average rate of change, Rolle’s theorem, the Mean Value Theorem and the Intermediate Value Theorem. (See 2007 AB 3 – four simple parts that could be multiple-choice questions; the mean on this question was 0.96 out of a possible 9.)
• These questions are usually presented in some context and answers should be in that context.
• Unit analysis.

Do’s and Don’ts

Do: Remember that you do not know what happens between the values in the table unless some other information is given. For example, don’t assume that the largest number in the table is the maximum value of the function.

Do: Show what you are doing even if you can do it in your head. If you’re finding a slope, show the quotient.

Do Not do arithmetic: A long expression consisting entire of numbers such as you get when doing a Riemann sum, does not need to be simplified in any way. If you simplify correct answer incorrectly, you will lose credit. However, do not leave expression such as R(3) – pull its numerical value from the table.

Do Not: Find a regression equation and then use that to answer parts of the question. While regression is perfectly good mathematics, regression equations are not one of the four things students may do with their calculator. Regression gives only an approximation of our function. The exam is testing whether students can work with numbers.

Shorter questions on this concept appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.

Next Posts:

Tuesday Match 21: Differential Equations (Type 6)

Friday March 24: Others (Type 7: related rates, implicit differentiation, etc.)

Tuesday March 28: for BC Parametric Equation (Type 8)

# Parts and More Parts

At an APSI this summer the participants and I got to discussing the “tabular method” for integration by parts. Since we were getting far from what is tested on the BC Calculus exams, I ended the discussion and said for those that were interested I would post more on the tabular method this blog going farther than just the basic set up. So here goes.

Here are some previous posts on integration by parts and the tabular method

Integration by Parts 1 discusses the basics of the method. This is as far as a BC course needs to go.

Integration by Parts 2 introduces the tabular method

Modified Tabular Integration presents a very quick and slick way of doing the tabular method without making a table. This is worth knowing.

There is also a video on integration by parts here. Scroll down to “Antiderivatives 5: A BC topic – Integration by parts.” The tabular method is discussed starting about time 15:16. There are several ways of setting up the table; one is shown here and a slightly different way is in the Integration by Parts 2 post above. There are others.

### Going further with the tabular method.

The tabular method works well if one of the factors in the original integrand is a polynomial; eventually its derivative will be zero and you are done. These are shown in the examples in the posts above and Example 1 below. To complete the topic, this post will show two other things that can happen when using integration by parts and the tabular method.

First we look at an example with a polynomial factor and learn how to stop midway through. Why stop? Because often there will be no end if you don’t stop. There are ways to complete the integration as shown in the examples.

Example 1:  Find $\displaystyle \int_{{}}^{{}}{\left( 4{{x}^{3}} \right)\cos \left( x \right)dx}$ by the tabular method (See Integration by Parts 2 for more detail on how to set the table up)

Adding the last column gives the antiderivative:

$\displaystyle \int_{{}}^{{}}{\left( 4{{x}^{3}} \right)\cos \left( x \right)dx}=4{{x}^{3}}\sin \left( x \right)+12{{x}^{2}}\cos \left( x \right)-24x\sin \left( x \right)-24\sin \left( x \right)+C$

Now say you wanted to stop after $12{{x}^{2}}\cos \left( x \right)$. Example 2 shows why you want (need) to stop. In Example 1 you will have

$\displaystyle \int_{{}}^{{}}{\left( 4{{x}^{3}} \right)\cos \left( x \right)dx}=4{{x}^{3}}\sin \left( x \right)+12{{x}^{2}}\sin \left( x \right)+\int_{{}}^{{}}{-24x\cos \left( x \right)}dx$

The integrand on the right is the product of the last column in the row at which you stopped and the first two columns in the next row, as shown in yellow above.

Example 2 Find $\displaystyle \int_{{}}^{{}}{{{e}^{x}}\cos \left( x \right)dx}$

As you can see things are just repeating the lines above sometimes with minus signs. However, if we stop on the third line we can write:

$\displaystyle \int_{{}}^{{}}{{{e}^{x}}\cos \left( x \right)dx={{e}^{x}}\sin \left( x \right)}+{{e}^{x}}\cos \left( x \right)-\int_{{}}^{{}}{{{e}^{x}}\cos \left( x \right)dx}$

The integral at the end is identical to the original integral.  We can continue by adding the integral to both sides:

$\displaystyle 2\int_{{}}^{{}}{{{e}^{x}}\cos \left( x \right)dx={{e}^{x}}\sin \left( x \right)}+{{e}^{x}}\cos \left( x \right)$

Finally, we divide by 2 and have the antiderivative we were trying to find:

$\displaystyle \int_{{}}^{{}}{{{e}^{x}}\cos \left( x \right)dx=\tfrac{1}{2}{{e}^{x}}\sin \left( x \right)}+\tfrac{1}{2}{{e}^{x}}\cos \left( x \right)+C$

In working this type of problem you must be aware of that the original integrand showing up again can happen and what to do if it does. As long as the coefficient is not +1, we can proceed as above. The same thing happens if we do not use the tabular method. (If the coefficient is +1 then the other terms on the right will add to zero and you need to make different choices for u and dv.)

### Reduction Formulas.

Another use of integration by parts is to produce formulas for integrals involving powers. An integral whose integrand is of less degree than the original, but of the same form results. The formula is then iterated to continually reduce the degree until the final integral can be integrated easily.

Example 3: Find $\displaystyle \int_{{}}^{{}}{{{x}^{n}}{{e}^{x}}dx}$

Let $u={{x}^{n}},\ du=n{{x}^{n-1}}dx,\ dv={{e}^{x}}dx,\ v={{e}^{x}}$

$\displaystyle \int_{{}}^{{}}{{{x}^{n}}{{e}^{x}}dx}={{x}^{n}}{{e}^{x}}-n\int_{{}}^{{}}{{{x}^{n-1}}{{e}^{x}}dx}$

This is a reduction formula; the second integral is the same as the first, but of lower degree. Here is how it is used.  At each step the integrand is the same as the original, but one degree lower. So the formula can be applied again, three more times in this example.

Most textbooks have a short selection of reduction formulas.

### Final Thoughts.

Back in the “old days”, BC (before calculators), beginning calculus courses spent a lot of time on the topic of “Techniques of Integration.” This included integration by parts, algebraic techniques, techniques known as trig-substitutions, and others. Mathematicians and engineers had tables of integrals listing over a thousand forms and students were taught how to use the tables and distinguish between similar forms in the tables. (See the photo below from the fourteenth edition of the CRC tables (c) 1965.) Current textbooks often contain such sections still.

Today, none of this is necessary. CAS calculators can find the antiderivatives of almost any integral. Websites such as WolframAlpha are also available to do this work.

I’m not sure why the College Board recently expanded slightly the list of types of antiderivatives tested on the exams. Certainly a few of the basic types should be included in a course, but what students really need to know is how to write the integral appropriate to a problem, and what definite and indefinite integrals mean. This, in my opinion, is far more important than being able to crank out antiderivatives of increasingly complicated expressions: let technology do that – or buy yourself an integral table. Just saying … .

# Modified Tabular Integration

Several weeks ago, Dr. Qibo Jing, an AP teacher at Rancho Solano Preparatory School in Scottsdale, Arizona, posted a new way to approach tabular integration to the AP Calculus Community discussion group.  He calls this the Modified Tabular Method. The algorithm makes repeated integration by parts quicker and more streamlined than the usual method. The usual method is explained here.

Here is Dr. Jing’s method outlined and illustrated with this example $\displaystyle I=\int{\left( {{x}^{3}}+7{{x}^{2}} \right)\cos \left( x \right)dx}$

Step 1: Identify u and dv in the usual way and rewrite the given integral in terms of $u={{x}^{3}}+7{{x}^{2}}$ and $dv=\cos \left( x \right)=\tfrac{d}{dx}\sin \left( x \right)$.

$\displaystyle I=\int{\left( {{x}^{3}}+7{{x}^{2}} \right)\cos \left( x \right)dx}=\int{\left( {{x}^{3}}+7{{x}^{2}} \right)\left( \tfrac{d}{dx}\left( \sin x \right) \right)}$

Step 2: The first term of the antiderivative is the product of the two functions that appear in the revised integral:

$\displaystyle I=\int{\left( {{x}^{3}}+7{{x}^{2}} \right)\left( \tfrac{d}{dx}\left( \sin x \right) \right)}=\left( {{x}^{3}}+7{{x}^{2}} \right)\sin \left( x \right)\cdots$

Step 3: The remaining terms alternate sign. The next term has subtraction sign etc. The first factor of the next term is the derivative of the first factor of the current term. The second factor is the antiderivative of the second factor of the current term – differentiate the first factor; integrate the second factor.

$\displaystyle I =\int{\left( {{x}^{3}}+7{{x}^{2}} \right)\left( \tfrac{d}{dx}\left( \sin x \right) \right)}=\left( {{x}^{3}}+7{{x}^{2}} \right)\sin \left( x \right)-\left( 3{{x}^{2}}+14x \right)\left( -\cos \left( x \right) \right)\cdots$

Step 4: Alternate signs and repeat step 3 until the term becomes zero or until the original form appears (See next example). DO NOT simplify either factor until you are done as this may change later terms.

The simplified result is

$\displaystyle I=\left( {{x}^{3}}+7{{x}^{2}}-6x-14 \right)\sin \left( x \right)+\left( 3{{x}^{2}}+14x-6 \right)\cos \left( x \right)+C$

Here is a second example.  In this example the original integral returns as the third term.

$\displaystyle I=\int{{{e}^{x}}\sin \left( x \right)dx=\int{\sin \left( x \right)\left( \tfrac{d}{dx}{{e}^{x}}dx \right)}}$

$=\sin \left( x \right)\left( {{e}^{x}} \right)-\cos \left( x \right)\left( {{e}^{x}} \right)+\left( -\sin \left( x \right) \right)\left( {{e}^{x}} \right)$

$={{e}^{x}}\sin \left( x \right)-\cos \left( x \right)\left( {{e}^{x}} \right)-I$

$2I={{e}^{x}}\sin \left( x \right)-\cos \left( x \right)\left( {{e}^{x}} \right)$

$I=\tfrac{1}{2}\left( {{e}^{x}}\sin \left( x \right)-\cos \left( x \right)\left( {{e}^{x}} \right) \right)+C$

This really is a simpler algorithm than the usual tabular method.