Tag Archives: Riemann sums

Why the Fundamental Theorem of Calculus?

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The Fundamental Theorem of Calculus is, well, fundamental. It relates the derivative and the integral.

Writing a Riemann sum with all that fancy notation is tedious. To speed things up a special notation is used to replace it. The limit of the Riemann sum for a function on an interval [a, b] is written as its definite integral:

\displaystyle \underset{{\left| {\Delta x} \right|\to 0}}{\mathop{{\lim }}}\,\sum\limits_{{i=1}}^{\infty }{{f\left( {{{x}_{i}}} \right)\Delta x}} \displaystyle =\int_{a}^{b}{{f\left( x \right)dx}}

The \displaystyle f\left( x \right) (called the integrand) is the function with no fancy notation and the dx, called differential x replaces the \displaystyle \Delta x. The a and b, called the lower and upper limit of integration respectively, show you the interval the Riemann sum was formed on (which the Riemann sum does not).

Keep in mind that behind every definite integral is a Riemann sum. Therefore, all the properties of limits apply to definite integrals. They can be added and subtracted, a constant may be factored out, and so on.

The Fundamental Theorem of Calculus, the FTC, tells you how to evaluate a definite integral (and therefore its Riemann sum): Simply evaluate the function of which \displaystyle f\left( x \right) is the derivative at the endpoints of the interval and subtract.

To keep this in mind you can write the FTC like this considering the integrand as the derivative (of something):

\displaystyle \int_{a}^{b}{{{f}'\left( x \right)dx}}=f\left( b \right)-f\left( a \right).

For example, since  \displaystyle d\sin \left( x \right)=\cos \left( x \right)dx,

\displaystyle \int_{0}^{{\pi /2}}{{\cos \left( x \right)dx=\sin }}\left( {\tfrac{\pi }{2}} \right)-\sin \left( 0 \right)=1-0=1

That’s all there is to it!

But wait! There’s more! This reveals another important idea: Since derivatives are rates of change, the FTC says that the integral of a rate of change is the net amount of change over the interval. Also called the accumulated change.

Well, okay, there is the problem of finding the function whose derivative is the integrand which is not always easy. This function is called the antiderivative of the integrand; another name is the indefinite integral. (The notation for an antiderivative or indefinite integral is the same as for a definite integral without the limits of integration). The truth is that finding the antiderivative is not as straightforward as finding the derivative. We will tackle that soon.

Course and Exam Description Unit 6.3 thru 6.7

Riemann Sum & Table Problems (Type 5)

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AP Questions Type 5: Riemann Sum & Table Problems

Information given in 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 are part of the Rule of Four and table problems are one way they are 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 and even if the denominator is 1.
  • Use a left-, right-, or midpoint- Riemann sums 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 nine points.)
  • These questions are usually presented in context and answers should be in that context. The context may be something growing (changing over time) or linear motion.
  • Use the table to find a value based on the Mean Value Theorem (2018 AB 4(b)) or Intermediate Value Theorem. Also, 2018 AB 4 (d) asked a related question based on a function given by an equation.
  • Unit analysis.

Dos and Don’ts

Do: Remember that you do not know what happens between the values in the table unless additional 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 between two values 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 even if the denominator is 1.

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 simplify a 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.

This question typically covers topics from Unit 6 of the CED but may include topics from Units 2, 3, and 4 as well.

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
  • 2018 AB 4
  • 2021 AB 1/ BC 1
  • 2022 AB4/BC4 – average rate of change, IVT, Rieman sum, Related Rate (part (d) good question)
  • 2023 AB 1/ BC 1 – Right Riemann sum, slope equals zero (may be done by max/min analysis, MVT, or Rolle’s Theorem), Equation extension – average value, meaning of derivative.

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)

Revised March 12, 2021, March 25, 2022, June 6, 2023

Riemann Sum & Table Problems (Type 5)

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AP  Questions Type 5: Riemann Sum & 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 and even if  the denominator is 1.
  • Use a left-, right-, or midpoint- Riemann sums 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. The context may be something growing (changing over time) or linear motion.
  • Use the table to find a value based on the Mean Value Theorem (2018 AB 4(b)) or Intermediate Value Theorem.
  • One of the parts of this question asks a related question based on a function given by an equation.
  • 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 between two values 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 even if the denominator is 1.

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.

This question typically covers topics from Unit 6 of the 2019 CED but may include topics from Units 2, 3, and 4 as well.

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
  • 2018 AB 4

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)

 

 

 

Revised March 12, 2021

 

 

2019 CED Unit 6: Integration and Accumulation of Change

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Unit 6 develops the ideas behind integration, the Fundamental Theorem of Calculus, and Accumulation. (CED – 2019 p. 109 – 128 ). These topics account for about 17 – 20% of questions on the AB exam and 17 – 20% of the BC questions.

Topics 6.1 – 6.4 Working up to the FTC

Topic 6.1 Exploring Accumulations of Change Accumulation is introduced through finding the area between the graph of a function and the x-axis. Positive and negative rates of change, unit analysis.

Topic 6.2 Approximating Areas with Riemann Sums Left-, right-, midpoint Riemann sums, and Trapezoidal sums, with uniform partitions are developed. Approximating with numerical methods, including use of technology are considered. Determining if the approximation is an over- or under-approximation.

Topic 6.3 Riemann Sums, Summation Notation and the Definite Integral. The definition integral is defined as the limit of a Riemann sum.

Topic 6.4 The Fundamental Theorem of Calculus (FTC) and Accumulation Functions Functions defined by definite integrals and the FTC.

Topic 6.5 Interpreting the Behavior of Accumulation Functions Involving Area Graphical, numerical, analytical, and verbal representations.

Topic 6.6 Applying Properties of Definite Integrals Using the properties to ease evaluation, evaluating by geometry and dealing with discontinuities.

Topic 6.7 The Fundamental Theorem of Calculus and Definite Integrals Antiderivatives. (Note: I suggest writing the FTC in this form \displaystyle \int_{a}^{b}{{{f}'\left( x \right)}}dx=f\left( b \right)-f\left( a \right) because it seems more efficient than using upper case and lower-case f.)

Topics 6.5 – 6.14 Techniques of Integration

Topic 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation. Using basic differentiation formulas to find antiderivatives. Some functions do not have closed-form antiderivatives. (Note: While textbooks often consider antidifferentiation before any work with integration, this seems like the place to introduce them. After learning the FTC students have a reason to find antiderivatives. See Integration Itinerary

Topic 6.9 Integration Using Substitution The u-substitution method. Changing the limits of integration when substituting.

Topic 6.10 Integrating Functions Using Long Division and Completing the Square 

Topic 6.11 Integrating Using Integration by Parts (BC ONLY)

Topic 6.12 Integrating Using Linear Partial Fractions (BC ONLY)

Topic 6.13 Evaluating Improper Integrals (BC ONLY) Showing the work requires students to show correct limit notation.

Topic 6.14 Selecting Techniques for Antidifferentiation This means practice, practice, practice.

Timing

The suggested time for Unit 6 is  18 – 20 classes for AB and 15 – 16 for BC of 40 – 50-minute class periods, this includes time for testing etc.

Previous posts on these topics include:

Introducing the Derivative

Integration Itinerary

The Old Pump and Flying to Integrationland   Two introductory explorations

Working Towards Riemann Sums

Riemann Sums

The Definition of the Definite Integral

Foreshadowing the FTC 

The Fundamental Theorem of Calculus

More About the FTC

Y the FTC?

Area Between Curves

Under is a Long Way Down 

Properties of Integrals 

Trapezoids – Ancient and Modern  On Trapezoid sums

Good Question 9 – Riemann Reversed   Given a Riemann sum can you find the Integral it converges to?  A common and difficult AP Exam problem

Accumulation

Accumulation: Need an Amount?

Good Question 7 – 2009 AB 3

Good Question 8 – or Not?  Unit analysis

AP Exams Accumulation Question    A summary of accumulation ideas.

Graphing with Accumulation 1

Graphing with Accumulation 2

Accumulation and Differential Equations 

Painting a Point

Techniques of Integrations (AB and BC)

Antidifferentiation

Why Muss with the “+C”?

Good Question 13  More than one way to skin a cat.

Integration by Parts – a BC Topic

Integration by Parts 1

Integration by Part 2

Parts and More Parts

Good Question 12 – Parts with a Constant?

Modified Tabular Integration 

Improper Integrals and Proper Areas

Math vs the Real World Why \displaystyle \int_{{-\infty }}^{\infty }{{\frac{1}{x}}}dx does not converge.

Here are links to the full list of posts discussing the ten units in the 2019 Course and Exam Description.

2019 CED – Unit 1: Limits and Continuity

2019 CED – Unit 2: Differentiation: Definition and Fundamental Properties.

2019 CED – Unit 3: Differentiation: Composite , Implicit, and Inverse Functions

2019 CED – Unit 4 Contextual Applications of the Derivative  Consider teaching Unit 5 before Unit 4

2019 – CED Unit 5 Analytical Applications of Differentiation  Consider teaching Unit 5 before Unit 4

2019 – CED Unit 6 Integration and Accumulation of Change

2019 – CED Unit 7 Differential Equations  Consider teaching after Unit 8

2019 – CED Unit 8 Applications of Integration   Consider teaching after Unit 6, before Unit 7

2019 – CED Unit 9 Parametric Equations, Polar Coordinates, and Vector-Values Functions 

2019 CED Unit 10 Infinite Sequences and Series

Riemann Sums – the Theory

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The series of post leads up to the Fundamental theorem of Calculus (FTC). Obviously, a very important destination.

  1. Working Towards Riemann Sums
  2. Definition of the Definite Integral and the FTC – a more exact demonstration from last Friday’s post and The Fundamental Theorem of Calculus –  an older demonstration
  3. More about the FTC The derivative of a function defined by an integral – the other half of the FTC.
  4. Good Question 11 Riemann Reversed – How to find the integral, given the Riemann sum. A problem that appears on the AP Calculus exams and can be confusing for students, at first.
  5. Properties of Integrals
  6. Variation on a Theme – 2 Comparing Riemann sums
  7. Trapezoids – Ancient and Modern – some history.

 

 

 

 

Revised and updated October 22, 2018

Getting Ready to Integrate

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Behind every definite integral is a Riemann sums. Students need to know about Riemann sums so that they can understand definite integrals (a shorthand notation for the limit if a Riemann sum) and the Fundamental theorem of Calculus. Theses posts help prepare students for Riemann sums.

Integration Itinerary  Some thoughts on the order of topics in your integration unit.

Some preliminary posts leading up to Riemann sums

  1. The Old Pump Where I start Integration
  2. Flying into Integrationland Continues the investigation in the Old Pump – the airplane problem
  3. Jobs, Jobs, Jobs Integration in real life.
  4. Working Towards Riemann Sums (12-10-2012)

While I prefer to teach antidifferentiation after students have learned about the Fundamental Theorem of Calculus, others prefer to discuss antidifferentiation firsts and the topic often precedes Riemann sums in textbooks. (See Integration Itinerary )  If you are among those, here are posts on antidifferentiation. If you teach this topic later, save this post for then.

ANTIDIFFERENTIATION

Antidifferentiation  (11-28-2012)

Why Muss with the “+C”? But still don’t forget it.

Arbitrary Ranges (2-9-2014) Integrating inverse trigonometric functions.

ANTIDIFFERENTIATION BY PARTS This is a BC topic, or you could use it after the exam in an AB course.

Integration by Parts 1 (2-2-2013) Basics

Integration by Parts 2 (2-4-2013) The Tabular Method

Modified Tabular Integration (7-24-2013) A quicker way

Parts and More Parts (8-5-2016) Reduction formulas (Not tested on the AP Calculus exams)

Good Question 12 – Parts with a Constant (12-13-2016) How come you don’t need the “+C”?

 

 

 

 

Trapezoids – Ancient and Modern

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The other day, in the course of about 10 minutes, I came across two interesting things about Trapezoidal approximations that I thought I would share with you.

cuneifirm tablet

Cuneiform writing

The first was a link to a story about how the ancient Babylonian astronomers sometime between 350 and 50 BCE used trapezoids to, in effect, find the area under a velocity-time graph tracking Jupiter’s motion. This was an NPR story based on a January 2016 Science magazine article in which the author, Mathieu Ossendrijver discusses his work deciphering cuneiform tablets written over 1,400 years before the technique showed up in Europe.

The second was a question asked on the AP Calculus bulletin board. A teacher asked, “Can someone please help me answer this question a student posed the other day. We were comparing left, right and midpoint and trapezoidal approximations. He asked since the trapezoidal calculation is the best estimate what is the use of LRAM and RRAM?” Here is an expanded form of my answer.

There are several things to consider here.

  1. First, if all you need is an estimate of the area or integral of a continuous function then a Trapezoid sum is certainly better than the left Riemann sum (left-RΣ) or the right-RΣ. Better, yes, the “best” maybe not: midpoint sums are about as good and parabola sums (Simpson’s Rule) are better.

2. Another reason to do left RΣ and right RΣ with small values of n is simply to give students practice in setting up Riemann sums so that they will be familiar with them when they move on to finding their limits and getting ready to define definite integrals.

3. A RΣ for a function f on a closed interval [a, b] is formed by partitioning the interval into subintervals and taking exactly one function value from each closed subinterval, multiplying the value by the width of that subinterval and adding these results. You may pick the function value any way you want – left end, middle, right end, any place at random in the subintervals and someplace else in the next subinterval. One way is to pick the smallest function value in each subinterval; this gives a RΣ called the lower RΣ. Likewise, you could pick the largest value in each subinterval; this gives the upper RΣ. Now it is true that

lower RΣ ≤ (any/all other RΣs) ≤ upper RΣ

Then as you add more partition points (n approaches infinity, or Δx approaches 0, etc.) the lower sum increases and the upper sum decreases. The series of lower sums is increasing and bounded above (by the upper sum) and therefore converges to its least upper bound. The upper sum decreases forming a decreasing series that is bounded below and therefore converges to its greatest lower bound.

If the lower RΣ and the upper RΣ approach the same value, then ALL the other RΣs approach that same value by the Squeeze theorem. This value is then defined as the definite integral of f from a to b.

In most AP calculus course, the textbooks do not deal with upper and lower sums. Instead, they deal with left RΣ and right RΣ on intervals on which f is only increasing (or only decreasing). In this case the lower RΣ = left RΣ and the upper RΣ = the right RΣ (or the other way around for decreasing functions).

So, this is why you need the left RΣ and right RΣ; not so much to approximate, but to complete the theory leading to the definite integral.