Area and Volume Problems (Type 4)

AP Type Questions 4: Area and Volume

Given equations that define a region in the plane students are asked to find its area, the volume of the solid formed when the region is revolved around a line, and/or the region is used as a base of a solid with regular cross-sections. This standard application of the integral has appeared every year since year one (1969) on the AB exam and almost every year on the BC exam. You can be fairly sure that if a free-response question on areas and volumes does not appear, the topic will be tested on the multiple-choice section.

What students should be able to do:

  • Find the intersection(s) of the graphs and use them as limits of integration (calculator equation solving). Write the equation followed by the solution; showing work is not required. Usually, no credit is earned until the solution is used in context (e.g., as a limit of integration). Students should know how to store and recall these values to save time and avoid copy errors.
  • Find the area of the region between the graph and the x-axis or between two graphs.
  • Find the volume when the region is revolved around a line, not necessarily an axis or an edge of the region, by the disk/washer method. See “Subtract the Hole from the Whole”
  • The cylindrical shell method will never be necessary for a question on the AP exams but is eligible for full credit if properly used.
  • Find the volume of a solid with regular cross-sections whose base is the region between the curves. For an interesting variation on this idea see 2009 AB 4(b)
  • Find the equation of a vertical line that divides the region in half (area or volume). This involves setting up an integral equation where the limit is the variable for which the equation is solved.
  • For BC only – find the area of a region bounded by polar curves: \displaystyle A=\frac{1}{2}{{\int_{{{{\theta }_{1}}}}^{{{{\theta }_{2}}}}{{\left( {r\left( \theta \right)} \right)}}}^{2}}d\theta
  • For BC only – Find perimeter using arc length integral

If this question appears on the calculator active section, it is expected that the definite integrals will be evaluated on a calculator. Students should write the definite integral with limits on their paper and put its value after it. It is not required to give the antiderivative and if a student gives an incorrect antiderivative, they will lose credit even if the final answer is (somehow) correct.

There is a calculator program available that will give the set-up and not just the answer so recently this question has been on the no calculator allowed section. (The good news is that in this case the integrals will be easy, or they will be set-up-but-do-not-integrate questions.)

Occasionally, other type questions have been included as a part of this question. See 2016 AB5/BC5 which included an average value question and a related rate question along with finding the volume.

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.

For some previous posts on this subject see January 911, 2013 and “Subtract the Hole from the Whole” of December 6, 2016.


The Area and Volume question covers topics from Unit 6 of the CED .


Free-response questions:

  • Variations: 2009 AB 4, Don’t overlook this one, especially part (b)
  • 2016 AB5/BC5,
  • 2017 AB 1 (using a table),
  • 2018 AB 5 – average rate of change, L’Hospital’s Rule
  • 2019 AB 5
  • Perimeter parametric curves 2011 BC 3 and 2014 BC 5
  • Area in polar form 2017 BC 5, 2018 BC 5, 20129 BC 2
  • 2021 AB 4/ BC 4
  • 2022 AB2 – area, volume, inc/dec analysis, and related rate.

Multiple-choice questions from non-secure exams:

  • 2008 AB 83 (Use absolute value),
  • 2012 AB 10, 92
  • 2012 BC 87, 92 (Polar area)

Revised March 12, 2012, March 22, 2022


Unlimited

Or when is a limit not a limit?

I was discussing the definition of a limit equal to infinity with someone recently. It occurred to me that such functions have no limit! Of course, you say that’s why – sometimes – we say “infinity”. But should we? What does “∞” mean?

The definition we were discussing is this:

\underset{{x\to a}}{\mathop{{\lim }}}\,f\left( x \right)=\infty \text{ if, and only if, for any }M>0\text{ there is a number }\delta >0\text{ such that} \text{ if }\left| {x-a} \right|<\delta \text{, then }f(x)>M.

What is being defined here?

What this definition says is that if we can always find numbers close to x = a that make the function’s value larger than any (every, all) positive number we pick, then we say that the limit is (equal to) infinity (∞).

This is how we say mathematically that every (any, all) number in the open interval defined by \left| {x=a} \right|<\delta , also known as a-\delta <x<a+\delta , will generate function values greater than M.

\displaystyle f\left( x \right)=\frac{1}{{{{{\left( {x-2} \right)}}^{2}}}}

For example, if \displaystyle f\left( x \right)=\frac{1}{{{{{\left( {x-2} \right)}}^{2}}}}, then \displaystyle \delta =\frac{1}{{\sqrt{M}}} will do the trick, since if

\displaystyle 0<\left| {x-2} \right|<\frac{1}{{\sqrt{M}}}

\displaystyle \Rightarrow \sqrt{M}>\left| {\frac{1}{{x-2}}} \right|

\displaystyle \Rightarrow M>\frac{1}{{{{{\left( {x-2} \right)}}^{2}}}}

The function has no value at exactly x = 2. As x get closer to 2, the graph just goes up and up; the values will eventually be greater than any value you choose. The line, x = 2 that the graph never gets to is called an asymptote. An asymptote is the graphical manifestation of a limit of infinity.

But wait a minute: this function has no limit; the values are unlimited. They just get larger and larger. It is correct to say, \underset{{x\to 2}}{\mathop{{\lim }}}\,\frac{1}{{{{{\left( {x-2} \right)}}^{2}}}}\text{ does not exist}. So, which is it “infinity” or does not exist”?

What is this ∞ thing?

An equal sign means that the numbers on both sides are the same. Now the limit part should be a number, but ∞ is not. So how can they be the same?

Is this an abuse of notation?

A definition must be phrased using previously defined terms. Have we defined ∞?

Up to the beginning of the calculus, we probably told students that infinity means something is greater than any value you choose. That’s true, but not much of a definition. (I hope you did not say “a number greater that any number you choose” or “the largest number,” because infinity is not a number.)

How can we tell if something is greater than any value we choose? The answer is that is exactly what the definition quoted above says! It defines what to say about situation where a function’s values get greater and greater, where they are unlimited. In doing so, it defines infinity as much as anything else, and maybe more so.


Disclaimer: There are functions whose limits fail to exist for other reasons and that “infinity” is not an appropriate description in those situations.

For example,  \displaystyle \underset{{x\to 3}}{\mathop{{\lim }}}\,\frac{{{{x}^{2}}-9}}{{\left( {{{x}^{2}}+1} \right)\sqrt{{{{{\left( {x-3} \right)}}^{2}}}}}} does not exist and is not ∞ .

\displaystyle f(x)=\frac{{{{x}^{2}}-9}}{{\left( {{{x}^{2}}+1} \right)\sqrt{{{{{\left( {x-3} \right)}}^{2}}}}}} has a finite jump discontinuity at x = 3.


Unit 1 – Limits and Continuity

This is a re-post and update of the first in a series of posts from last year. It contains links to posts on this blog about the topics of limits and continuity for your reference in planning. Other updated post on the 2019 CED will come throughout the year, hopefully, a few weeks before you get to the topic. 

Unit 1 contains topics on Limits and Continuity. (CED – 2019 p. 36 – 50). These topics account for about 10 – 12% of questions on the AB exam and 4 – 7% of the BC questions.

Logically, limits come before continuity since limit is used to define continuity. Practically and historically, continuity comes first. Newton and Leibnitz did not have the concept of limit the way we use it today. It was in the early 1800’s that the epsilon-delta definition of limit was first given by Bolzano (whose work was overlooked) and then by Cauchy, Jordan, and Weierstrass. But, their formulation did not use the word “limit”, rather the use was part of their definition of continuity. Only later was it pulled out as a separate concept and then returned to the definition of continuity as a previously defined term. See Which Came First?

Students should have plenty of experience in their math courses before calculus with functions that are and are not continuous. They should know the names of the types of discontinuities – jump, removable, infinite, oscillating etc.and the related terms such as asymptote. As you go through this unit, you may want to quickly review these terms and concepts as they come up.

(As a general technique, rather than starting the year with a week or three of review – which the students need but will immediately forget again – be ready to review topics as they come up during the year as they are needed – you will have to do that anyway. See Getting Started #2)

Topics 1.1 – 1.9: Limits

Topic 1.1: Suggests an introduction to calculus to give students a hint of what’s coming. See Getting Started #3

Topic 21.: Proper notation and multiple-representations of limits.

There is an exclusion statement noting that the delta-epsilon definition of limit is not tested on the exams, but you may include it if you wish. The epsilon-delta definition is not tested probably because it is too difficult to write good questions. Specifically, (1) the relationship for a linear function is always  delta =frac{varepsilon }{{left| m right|}}  where m is the slope and is too complicated to compute for other functions, and (2) for a multiple-choice question the smallest answer must be correct. (Why?)

Topic 1.3: One-sided limits.

Topic 1.4: Estimating limits numerically and from tables.

Topic 1.5: Algebraic properties of limits.

Topic 1.6: Simplifying expressions to find their limits. This can and should be done along with learning the other concepts and procedures in this unit.

Topic 1.7: Selecting the proper procedure for finding a limit. The first step is always to substitute the value into the limit. If this comes out to be number than that is the limit. If not, then some manipulation may be required. This can and should be done along with learning the other concepts and procedures in this unit.

Topic 1.8: The Squeeze Theorem is mainly used to determine  underset{{xto 0}}{mathop{{lim }}},frac{{sin left( x right)}}{x}=1 which in turn is used in finding the derivative of the sin(x). (See Why Radians?) Most of the other examples seem made up just for exercises and tests. (See 2019 AB 6(d)). Thus, important, but not too important.

Topic 1.9: Connecting multiple-representations of limit. This can and should be done along with learning the other concepts and procedures in this unit. Dominance, Topic 15, may be included here as well (EK LIM-2.D.5)

Topics 1.10 – 1.16 Continuity

Topic 1.10: Here you can review the different types of discontinuities with examples and graphs.

Topic 1.11: The definition of continuity. The EK statement does not seem to use the three-hypotheses definition. However, for the limit to exist and for f(c) to exist, they must be real numbers (i.e. not infinite). This is tested often on the exams, so students should have practice with verifying that (all three parts of) the hypothesis are met and including this in their answers.

Topic 1.12: Continuity on an interval and which Elementary Functions are continuous for all real numbers.

Topic 1.13: Removable discontinuities and handing piecewise – defined functions

Topic 1.14: Vertical asymptotes and unbounded functions. Here be sure to explain the difference between limits “equal to infinity” and limits that do not exist (DNE). See Good Question 5: 1998 AB2/BC2.

Topic 1.15: Limits at infinity, or end behavior of a function. Horizontal asymptotes are the graphical manifestation of limits at infinity or negative infinity. Dominance is included here as well (EK LIM-2.D.5)

Topic 1.16: The Intermediate Value Theorem (IVT) is a major and important result of a function being continuous. This is perhaps the first Existence Theorem students encounter, so be sure to stop and explain what an existence theorem is.


The suggested number of 40 – 50 minute class periods is 22 – 23 for AB and 13 – 14 for BC. This includes time for testing etc. If time seems to be a problem you can probably combine topics 3 – 5, topics 6 -7, topics 11 – 12. Topics 6, 7, and 9 are used with all the limit work.


There are three other important limits that will be coming in later Units:

The definition of the derivative in Unit 2, topics 1 and 2

L’Hospital’s Rule in Unit 4, topic 7

The definition of the definite integral in Unit 6, topic 3.

Posts on Continuity

CONTINUITY To help understand limits it is a good idea to look at functions that are not continuous. Historically and practically, continuity should come before limits. On the other hand, the definition of continuity requires knowing about limits. So, I list continuity first. The modern definition of limit was part of Weierstrass’ definition of continuity.

Which Came First? (7-28-2020)

Continuity (8-13-2012)

Continuity (8-21-2013) The definition of continuity.

Continuous Fun (10-13-2015) A fuller discussion of continuity and its definition

Right Answer – Wrong Question (9-4-2013) Is a function continuous even if it has a vertical asymptote?

Asymptotes (8-15-2012) The graphical manifestation of certain limits

Fun with Continuity (8-17-2012) the Diriclet function

Far Out! (10-31-2012) When the graph and dominance “disagree” From the Good Question series

Posts on Limits

Why Limits? (8-1-2012)

Deltas and Epsilons (8-3-2012) Why this topic is not tested on the AP Calculus Exams.

Finding Limits (8-4-2012) How to…

Dominance (8-8-2012) See limits at infinity

Determining the Indeterminate (12-6-2015) Investigating an indeterminate form from a differential equation. From the Good Question series.

Locally Linear L’Hôpital (5-31-2013) Demonstrating L’Hôpital’s Rule (a/k/a L’Hospital’s Rule)

L’Hôpital’s Rules the Graph (6-5-2013)

Unlimited


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

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


Adapting 2021 AB 4 / BC 4

Four of nine. Continuing the series started in the last three posts, this post looks at the AP Calculus 2021 exam question AB 4 / BC 4. The series considers each question with the aim of showing ways to use the question with your class as is, or by adapting and expanding it.  Like most of the AP Exam questions there is a lot more you can ask from the stem and a lot of other calculus you can discuss.

2021 AB 4 / BC 4

This is a Graph Analysis Problem (type 3) and contains topics from Units 2, 4, and 6 of the current Course and Exam Description. The things that are asked in these questions should be easy for the students, however each year the scores are low. This may be because some textbooks simply do not give students problems like this. Therefore, supplementing with graph analysis questions from past exams is necessary.

There are many additional questions that can be asked based on this stem and the stems of similar problems. Usually, the graph of the derivative is given, and students are asked questions about the graph of the function. See Reading the Derivative’s Graph.

Some years this question is given a context, such as the graph is the velocity of a moving particle. Occasionally there is no graph and an expression for the derivative or function is given.

Here is the 2021 AB 4 / BC 4 stem:

The first thing students should do when they see G\left( x \right)=\int_{0}^{x}{{f\left( t \right)}}dt is to write prominently on their answer page {G}'\left( x \right)=f\left( x \right) and \displaystyle {G}''\left( x \right)={f}'\left( t \right). While they may understand and use this, they must say it.

Part (a): Students were asked for the open intervals where the graph is concave up and to give a reason for their answer. (Asking for an open interval is to remove any concern about the endpoints being included or excluded, a place where textbooks differ. See Going Up.)

Discussion and ideas for adapting this question:

  • Using this or similar graphs go through each of these with your class until the answers and reasons become automatic. There are quite a few other things that may be asked here based on the derivative.
    • Where is the function increasing?
    • Decreasing?
    • Concave down, concave up?
    • Where are the local extreme values?
    • What are the local extreme values?
    • Where are the absolute extreme values?
    • What are the absolute extreme values?
  • There are also integration questions that may be asked, such as finding the value of the functions at various points, such as G(1) = 2 found by using the areas of the regions. Also, questions about the local extreme values and the absolute extreme value including their values. These questions are answered by finding the areas of the regions enclosed by the derivative’s graph and the x-axis. Parts (b) and (c) do some of this.
  • Choose different graphs, including one that has the derivative’s extreme value on the x­-axis. Ask what happens there.

Part (b): A new function is defined as the product of G(x) and f(x) and its derivative is to be found at a certain value of x. To use the product rule students must calculate the value of G(x) by using the area between f(x) and the x-­axis and the value of {f}'\left( x \right) by reading the slope of f(x) from the graph.

Discussion and ideas for adapting this question:

  • This is really practice using the product rule. Adapt the problem by making up functions using the quotient rule, the chain rule etc. Any combination of \displaystyle G,{G}',{G}'',f,{f}',\text{ or }{f}'' may be used. Before assigning your own problem, check that all the values can be found from the given graph.
  • Different values of x may be used.

Part (c): Students are asked to find a limit. The approach is to use L’Hospital’s Rule.

Discussion and ideas for adapting this question:

  • To use L’Hospital’s Rule, students must first show clearly on their paper that the limit of the numerator and denominator are both zero or +/- infinity. Saying the limit is equal to 0/0 is considered bad mathematics and will not earn this point. Each limit should be shown separately on the paper, before applying L’Hospital’s Rule.
  • Variations include a limit where L’Hospital’s Rule does not apply. The limit is found by substituting the values from the graph.
  • Another variation is to use a different expression where L’Hospital’s Rule applies, but still needs values read from the graph.

Part (d): The question asked to find the average rate of change (slope between the endpoints) on an interval and then determine if the Mean Value Theorem guarantees a place where \displaystyle {G}' equals this value. Students also must justify their answer.

Discussion and ideas for adapting this question:

  • To justify their answer students must check that the hypotheses of the MVT are met and say so in their answer.
  • Adapt by using a different interval where the MVT applies.
  • Adapt by using an interval where the MVT does not apply and (1) the conclusion is still true, or (b) where the conclusion is false.

Next week 2021 AB 5.

I would be happy to hear your ideas for other ways to use this questions. Please use the reply box below to share your ideas.


Limits and Continuity – Unit 1

This is a re-post and update of the first in a series of posts from last year. It contains links to posts on this blog about the topics of limits and continuity for your reference in planning. Other updated post on the 2019 CED will come throughout the year, hopefully, a few weeks before you get to the topic. 

Unit 1 contains topics on Limits and Continuity. (CED – 2019 p. 36 – 50). These topics account for about 10 – 12% of questions on the AB exam and 4 – 7% of the BC questions.

Logically, limits come before continuity since limit is used to define continuity. Practically and historically, continuity comes first. Newton and Leibnitz did not have the concept of limit the way we use it today. It was in the early 1800’s that the epsilon-delta definition of limit was first given by Bolzano (whose work was overlooked) and then by Cauchy, Jordan, and Weierstrass. But, their formulation did not use the word “limit”, rather the use was part of their definition of continuity. Only later was it pulled out as a separate concept and then returned to the definition of continuity as a previously defined term. See Which Came First?

Students should have plenty of experience in their math courses before calculus with functions that are and are not continuous. They should know the names of the types of discontinuities – jump, removable, infinite, oscillating etc.and the related terms such as asymptote. As you go through this unit, you may want to quickly review these terms and concepts as they come up.

(As a general technique, rather than starting the year with a week or three of review – which the students need but will immediately forget again – be ready to review topics as they come up during the year as they are needed – you will have to do that anyway. See Getting Started #2)

Topics 1.1 – 1.9: Limits

Topic 1.1: Suggests an introduction to calculus to give students a hint of what’s coming. See Getting Started #3

Topic 21.: Proper notation and multiple-representations of limits.

There is an exclusion statement noting that the delta-epsilon definition of limit is not tested on the exams, but you may include it if you wish. The epsilon-delta definition is not tested probably because it is too difficult to write good questions. Specifically, (1) the relationship for a linear function is always  \delta =\frac{\varepsilon }{{\left| m \right|}}  where m is the slope and is too complicated to compute for other functions, and (2) for a multiple-choice question the smallest answer must be correct. (Why?)

Topic 1.3: One-sided limits.

Topic 1.4: Estimating limits numerically and from tables.

Topic 1.5: Algebraic properties of limits.

Topic 1.6: Simplifying expressions to find their limits. This can and should be done along with learning the other concepts and procedures in this unit.

Topic 1.7: Selecting the proper procedure for finding a limit. The first step is always to substitute the value into the limit. If this comes out to be number than that is the limit. If not, then some manipulation may be required. This can and should be done along with learning the other concepts and procedures in this unit.

Topic 1.8: The Squeeze Theorem is mainly used to determine  \underset{{x\to 0}}{\mathop{{\lim }}}\,\frac{{\sin \left( x \right)}}{x}=1 which in turn is used in finding the derivative of the sin(x). (See Why Radians?) Most of the other examples seem made up just for exercises and tests. (See 2019 AB 6(d)). Thus, important, but not too important.

Topic 1.9: Connecting multiple-representations of limit. This can and should be done along with learning the other concepts and procedures in this unit. Dominance, Topic 15, may be included here as well (EK LIM-2.D.5)

Topics 1.10 – 1.16 Continuity

Topic 1.10: Here you can review the different types of discontinuities with examples and graphs.

Topic 1.11: The definition of continuity. The EK statement does not seem to use the three-hypotheses definition. However, for the limit to exist and for f(c) to exist, they must be real numbers (i.e. not infinite). This is tested often on the exams, so students should have practice with verifying that (all three parts of) the hypothesis are met and including this in their answers.

Topic 1.12: Continuity on an interval and which Elementary Functions are continuous for all real numbers.

Topic 1.13: Removable discontinuities and handing piecewise – defined functions

Topic 1.14: Vertical asymptotes and unbounded functions. Here be sure to explain the difference between limits “equal to infinity” and limits that do not exist (DNE). See Good Question 5: 1998 AB2/BC2.

Topic 1.15: Limits at infinity, or end behavior of a function. Horizontal asymptotes are the graphical manifestation of limits at infinity or negative infinity. Dominance is included here as well (EK LIM-2.D.5)

Topic 1.16: The Intermediate Value Theorem (IVT) is a major and important result of a function being continuous. This is perhaps the first Existence Theorem students encounter, so be sure to stop and explain what an existence theorem is.


The suggested number of 40 – 50 minute class periods is 22 – 23 for AB and 13 – 14 for BC. This includes time for testing etc. If time seems to be a problem you can probably combine topics 3 – 5, topics 6 -7, topics 11 – 12. Topics 6, 7, and 9 are used with all the limit work.


There are three other important limits that will be coming in later Units:

The definition of the derivative in Unit 2, topics 1 and 2

L’Hospital’s Rule in Unit 4, topic 7

The definition of the definite integral in Unit 6, topic 3.

Posts on Continuity

CONTINUITY To help understand limits it is a good idea to look at functions that are not continuous. Historically and practically, continuity should come before limits. On the other hand, the definition of continuity requires knowing about limits. So, I list continuity first. The modern definition of limit was part of Weierstrass’ definition of continuity.

Which Came First? (7-28-2020)

Continuity (8-13-2012)

Continuity (8-21-2013) The definition of continuity.

Continuous Fun (10-13-2015) A fuller discussion of continuity and its definition

Right Answer – Wrong Question (9-4-2013) Is a function continuous even if it has a vertical asymptote?

Asymptotes (8-15-2012) The graphical manifestation of certain limits

Fun with Continuity (8-17-2012) the Diriclet function

Far Out! (10-31-2012) When the graph and dominance “disagree” From the Good Question series

Posts on Limits

Why Limits? (8-1-2012)

Deltas and Epsilons (8-3-2012) Why this topic is not tested on the AP Calculus Exams.

Finding Limits (8-4-2012) How to…

Dominance (8-8-2012) See limits at infinity

Determining the Indeterminate (12-6-2015) Investigating an indeterminate form from a differential equation. From the Good Question series.

Locally Linear L’Hôpital (5-31-2013) Demonstrating L’Hôpital’s Rule (a/k/a L’Hospital’s Rule)

L’Hôpital’s Rules the Graph (6-5-2013)


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

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


 

 

 

 

 

 

Which Came First?

In one of my math classes – it may have been calculus – many decades ago, we started by determining what kind of functions we were going to study. A good part of the answer was continuous functions. Looking closely, you will find that almost all the theorems in beginning calculus require that the function be continuous on an interval as one of their hypotheses (The interval could be all Real numbers.) Later theorems require that the function be differentiable, but, as you will learn, if a function is differentiable, then it is continuous. So, calculus studies continuous functions (or those that are not continuous at only a few points).

A function is continuous on an interval, roughly speaking, you can draw its graph from one side of the interval to the other without taking the pencil off the paper. Thus, if a function has a hole, a vertical asymptote, a jump, or oscillates wildly it is not continuous. Continuity is first determined for a function at a point in it domain. Then this is extended to all the points in an interval.

Students come across functions that are not continuous long before they encounter calculus and limits. They see functions with asymptotes, jumps, and holes long before calculus. Discussing continuity gives a reason to talk about limits informally and how the idea of “getting closer to” a point works. This eventually leads to the idea of a limit and the need to define the term.

The definition of continuity at a point that is used most often is this:

A function f is continuous at x=a if, and only if, (1)  f\left( a \right) exists, (2)  \underset{{x\to a}}{\mathop{{\lim }}}\,f\left( x \right) exists, and (3)  \underset{{x\to a}}{\mathop{{\lim }}}\,f\left( x \right)=f\left( a \right).

The first two conditions are probably included to prevent beginning students from thinking that if the value and the limit are both “infinite” as in the case with some vertical asymptotes, then the function is continuous. In fact, the two things can only be equal if they are finite.

The definition of limit (which is not tested on either AP Calculus exam) states that

 \underset{{x\to a}}{\mathop{{\lim }}}\,f\left( x \right)=L, if, and only if, for every number \varepsilon >0 there exists a number  \delta >0 such that if \left| {x-a} \right|<\delta , then \left| {f\left( x \right)-L} \right|<\varepsilon .

It was in the early 1800’s that the epsilon-delta definition of limit was first given by Bolzano (whose work was overlooked) and then by Cauchy, Jordan, and Weierstrass..Historically, the definition of continuity was first given by Karl Weierstrass (1815 – 1897) and  Camille Jordan (1838 – 1922). Their definition is:

A Real valued function is continuous at  x=a, if and only if, for every number \varepsilon >0 there exists a number  \delta >0 such that if \left| {x-a} \right|<\delta , then \left| {f\left( x \right)-L} \right|<\varepsilon .

As you can see, the original definition is simply the modern definition of limit applied to the concept of continuity.

So, which came first, continuity or limits?

Calculus textbooks and the 2019 Course and Exam Description for AP Calculus’s first unit begins with limits (lessons 1.2 to 1.9) and then continuity (lessons 1.10 – 1.16). They are being logical: the concept of limit is needed to define continuity.

So, logically you need limits to talk about continuity. Practically, continuity, or lack thereof, comes first. Students should be familiar with continuous graphs and the types of discontinuities before they start calculus. The calculus course will formalize things and make the ideas precise using limits.

______________________________

Stretch your brain a bit: Almost all the functions you will study are continuous at all but a few (a finite number) of places. If that were not so, there would not be much calculus you could “do.” But, consider the Dirichlet function:

D\left( x \right)=\left\{ {\begin{array}{*{20}{c}} 0 & {\text{if }x\text{ is irrational}} \\ 1 & {\text{if }x\text{ is rational}} \end{array}} \right.

Since there are always rational numbers between any two irrational numbers, and irrational numbers between any to rational numbers, this function is not continuous anywhere! No point is adjacent to any other point.

And a little more stretch: Discuss the continuity at x = 1 of this function:

L\left( x \right)=\left\{ {\begin{array}{*{20}{c}} x & {\text{ if }x\text{ is irrational}} \\ 1 & {\text{if }x\text{ is rational}} \end{array}} \right.


Next Tuesday, I will begin posting the lists of references to blog posts about topics related to the units of the 2019 Course and Exam Description for AP Calculus beginning with Unit 1: Limits and Continuity.



Area and Volume Problems (Type 4)

AP Type Questions 4: Area and Volume

Given equations that define a region in the plane students are asked to find its area, the volume of the solid formed when the region is revolved around a line, and/or the region is used as a base of a solid with regular cross-sections. This standard application of the integral has appeared every year since year one (1969) on the AB exam and almost every year on the BC exam. You can be pretty sure that if a free-response question on areas and volumes does not appear, the topic will be tested on the multiple-choice section.

What students should be able to do:

  • Find the intersection(s) of the graphs and use them as limits of integration (calculator equation solving). Write the equation followed by the solution; showing work is not required. Usually no credit is earned until the solution is used in context (as a limit of integration). Students should know how to store and recall these values to save time and avoid copy errors.
  • Find the area of the region between the graph and the x-axis or between two graphs.
  • Find the volume when the region is revolved around a line, not necessarily an axis or an edge of the region, by the disk/washer method. See “Subtract the Hole from the Whole”
  • The cylindrical shell method will never be necessary for a question on the AP exams, but is eligible for full credit if properly used.
  • Find the volume of a solid with regular cross-sections whose base is the region between the curves. For an interesting variation on this idea see 2009 AB 4(b)
  • Find the equation of a vertical line that divides the region in half (area or volume). This involves setting up an integral equation where the limit is the variable for which the equation is solved.
  • For BC only – find the area of a region bounded by polar curves: A=\tfrac{1}{2}\int\limits_{{{\theta }_{1}}}^{{{\theta }_{2}}}{{{\left( r\left( \theta  \right) \right)}^{2}}}d\theta
  • For BC only – Find perimeter using arc length integral

If this question appears on the calculator active section, it is expected that the definite integrals will be evaluated on a calculator. Students should write the definite integral with limits on their paper and put its value after it. It is not required to give the antiderivative and if a student gives an incorrect antiderivative they will lose credit even if the final answer is (somehow) correct.

There is a calculator program available that will give the set-up and not just the answer so recently this question has been on the no calculator allowed section. (The good news is that in this case the integrals will be easy or they will be set-up-but-do-not-integrate questions.)

Occasionally, other type questions have been included as a part of this question. See 2016 AB5/BC5 which included an average value question and a related rate question along with finding the volume.

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.

For some previous posts on this subject see January 911, 2013 and “Subtract the Hole from the Whole” of December 6, 2016.


The Area and Volume question covers topics from Unit 6 of the 2019 CED .


Free-response questions:

  • Variations: 2009 AB 4, Don’t overlook this one, especially part (b)
  • 2016 AB5/BC5,
  • 2017 AB 1 (using a table),
  • 2018 AB 5 – average rate of change, L’Hospital’s Rule
  • 2019 AB 5
  • Perimeter parametric curves 2011 BC 3 and 2014 BC 5
  • Area in polar form 2017 BC 5, 2018 BC 5, 20129 BC 2

Multiple-choice questions from non-secure exams:

  • 2008 AB 83 (Use absolute value),
  • 2012 AB 10, 92
  • 2012 BC 87, 92 (Polar area)

 

 

 

 

 

 

Revised March 12, 2012