Teaching double integrals: why you should *always* sketch the region

The problem (from Larson’s Calculus, an Applied Approach, 10’th edition, Section 7.8, no. 18 in my paperback edition, no. 17 in the e-book edition) does not seem that unusual at a very quick glance:

if you have a hard time reading the image. AND, *if* you just blindly do the formal calculations:

which is what the text has as “the answer.”

But come on. We took a function that was negative in the first quadrant, integrated it entirely in the first quadrant (in standard order) and ended up with a positive number??? I don’t think so!

Indeed, if we perform which is far more believable.

So, we KNOW something is wrong. Now let’s attempt to sketch the region first:

Oops! Note: if we just used the quarter circle boundary we obtain

The 3-dimensional situation: we are limited by the graph of the function, the cylinder and the planes ; the plane is outside of this cylinder. (from here: the red is the graph of

Now think about what the “formal calculation” really calculated and wonder if it was just a coincidence that we got the absolute value of the integral taken over the rectangle

My teaching during the COVID-19 Pandemic

My university has moved to “online only” for the rest of the semester. I realize that most of us are in the same boat.
Fortunately, for now, I’ve got some academic freedom to make changes and I am taking a different approach than some.

Some appear to be wanting to keep things “as normal as possible.”

For me: the pandemic changes everything.

Yes, there are those on the beach in Florida. That isn’t most of my students; it could be some of them.

So, here is what will be different for me:
1) I am making exams “open book, open note” and take home: they get it and are given several days to do it and turn it back in, like a project.
Why? Fluid situations, living with a family, etc. might make it difficult to say “you HAVE to take it now…during period X.” This is NOT an online class that they signed up for.
Yes, it is possible that some cheat; that can’t be helped.

Also, studying will be difficult to do. So, getting a relatively long “designed as a programmed text” is, well, getting them to study WHILE DOING THE EXAM. No, it is not the same as “study to put it in your brain and then show you know it” at exam time. But I feel that this gets them to learn while under this stressful situation; they take time aside to look up and think about the material. The exam, in a way, is going through a test bank.

2) Previously, I thought of testing as serving two purposes: a) it encourages students to review and learn and b) distinguishing those with more knowledge from those with lesser knowledge. Now: tests are to get the students to learn..of course diligence will be rewarded. But who does well and who does not..those groups might change a little.

3) Quiz credit: I was able to sign up for webassign, and their quizzes will be “extra credit” to build on their existing grade. This is a “carrot only” approach.

4) Most of the lesson delivery will be a polished set of typeset notes with videos. My classes will be a combination of “live chat” with video where I will discuss said notes and give tips on how to do problems. I’ll have office hours ..some combination of zoom meetings which people can join and I’ll use e-mail to set up “off hours” meetings, either via chat or zoom, or even an exchange of e-mails.

We shall see how it works; I have a plan and think I can execute it, but I make no guarantee of the results.
Yes, there are polished online classes, but those are designed to be done deliberately. What we have here is something made up at the last minute for students who did NOT sign up for it and are living in an emergency situation.

An easy fact about least squares linear regression that I overlooked

The background: I was making notes about the ANOVA table for “least squares” linear regression and reviewing how to derive the “sum of squares” equality:

Total Sum of Squares = Sum of Squares Regression + Sum of Squares Error or…

If is the observed response, the sample mean of the responses, and are the responses predicted by the best fit line (simple linear regression here) then:

(where each sum is for the n observations. )

Now for each it is easy to see that but the equations still holds if when these terms are squared, provided you sum them up!

And it was going over the derivation of this that reminded me about an important fact about least squares that I had overlooked when I first presented it.

If you go in to the derivation and calculate:

Which equals and the proof is completed by showing that:

and that BOTH of these sums are zero.

But why?

Let’s go back to how the least squares equations were derived:

Given that

yields that . That is, under the least squares equations, the sum of the residuals is zero.

Now which yields that

That is, the sum of the residuals, weighted by the corresponding x values (inputs) is also zero. Note: this holds with multilinear regreassion as well.

Really, that is what the least squares process does: it sets the sum of the residuals and the sum of the weighted residuals equal to zero.

Yes, there is a linear algebra formulation of this.

Anyhow returning to our sum:

Now for the other term:

Now as it is a constant multiple of the sum of residuals and as it is a constant multiple of the weighted sum of residuals..weighted by the .

That was pretty easy, wasn’t it?

But the role that the basic least squares equations played in this derivation went right over my head!

An economist talks about graphs

Paul Krugman is a Nobel Laureate caliber economist (he won whatever they call the economics prize).
Here he discusses the utility of using a graph to understand an economic situation:

Brad DeLong asks a question about which of the various funny diagrams economists love should be taught in Econ 101. I say production possibilities yes, Edgeworth box no — which, strange to say, is how we deal with this issue in Krugman/Wells. But students who go on to major in economics should be exposed to the box — and those who go on to grad school really, really need to have seen it, and in general need more simple general-equilibrium analysis than, as far as I can tell, many of them get these days.

There was, clearly, a time when economics had too many pictures. But now, I suspect, it doesn’t have enough.

OK, this is partly a personal bias. My own mathematical intuition, and a lot of my economic intuition in general, is visual: I tend to start with a picture, then work out both the math and the verbal argument to make sense of that picture. (Sometimes I have to learn the math, as I did on target zones; the picture points me to the math I need.) I know that’s not true for everyone, but it’s true for a fair number of students, who should be given the chance to learn things that way.

Beyond that, pictures are often the best way to convey global insights about the economy — global in the sense of thinking about all possibilities as opposed to small changes, not as in theworldisflat. […]

And it probably doesn’t hurt to remind ourselves that our students are, in general, NOT like us. What comes to us naturally probably does not come to them naturally.

Where these posts are (often) coming from

Yes, my office is messy. Deal with it. 🙂 And yes, some of my professional friends (an accountant and a lawyer) just HAD to send me their office shots…pristine condition, of course.
(all in good fun!)

Note: this semester I teach 3 classes in a row: second semester “business/life science” calculus, second semester “engineering/physical science” calculus and linear algebra. Yes, I love the topics, but there is just enough overlap that I have to really clear my head between lessons. Example: we covered numerical integration in both of my calculus classes, as well as improper integrals. I have to be careful not to throw in as an example during my “life science calculus” class. I do the “head clearing” by going up the stairs to my office between classes.

Linear algebra is a bit tricky; we are so accustomed to taking things like “linear independence” for granted that it is easy to forget that this is the first time the students are seeing it. Also, the line between rigor and “computational usefulness” is tricky; for example, how rigorously do we explain “the determinant” of a matrix?

Oh well…back to some admin nonsense.

Topologist’s Sine Curve: connected but not path connected.

I wrote the following notes for elementary topology class here. Note: they know about metric spaces but not about general topological spaces; we just covered “connected sets”.

I’d like to make one concession to practicality (relatively speaking). When it comes to showing that a space is path connected, we need only show that, given any points there exists where is continuous and . Here is why: by maps to homeomorphically provided and so provides the required continuous function from into .

Now let us discuss the topologist’s sine curve. As usual, we use the standard metric in and the subspace topology.

Let . See the above figure for an illustration. is path connected as, given any two points in , then is the required continuous function . Therefore is connected as well.

Note that is a limit point for though .

Exercise: what other limit points does that are disjoint from ?

Now let , that is, we add in the point at the origin.

Fact: is connected. This follows from a result that we proved earlier but here is how a “from scratch” proof goes: if there were open sets in that separated in the subspace topology, every point of would have to lie in one of these, say because is connected. So the only point of that could lie in would be which is impossible, as every open set containing hits a point (actually, uncountably many) of .

Now we show that is NOT path connected. To do this, we show that there can be no continuous function where .

One should be patient with this proof. It will go in the following stages: first we show that any such function must include EVERY point of in its image and then we show that such a function cannot be extended to be continuous at .

First step: for every there exists where Suppose one point was missed; let denote the least upper bound of all coordinates of points that are not in the image of . By design (why: continuity and the fact that ) So cuts the image of TS into two disjoint open sets (in the subspace topology): that part with x-coordinate less than and that part with x-coordinate greater than . So and form separating open sets for which is impossible.

Note: if you don’t see the second open set in the picture, note that for all one can find and open disk that misses the part of the graph that occurs “before” the coordinate . The union of these open disks (an uncountable union) plus an open disk around forms ; remember that an arbitrary union of open sets is open.

Second step: Now we know that every point of is hit by . Now we can find the sequence and note that in . But we can also find where in . So we have two sequences in the domain converging to the same number but going to different values after applying . That is impossible if is continuous.

This gives us another classification result: and are not topologically equivalent as is not path connected.

How to succeed at calculus, and why it is worth it!

This post is intended to help the student who is willing to put time and effort into succeeding in a college calculus class.

Part One: How to Study

The first thing to remember is that most students will have to study outside of class in order to learn the material. There are those who pick things up right away, but these students tend to be the rare exception.

Think of it this way: suppose you want to learn to play the piano. A teacher can help show you how to play it and provide a practice schedule. But you won’t be any good if you don’t practice.

Suppose you want to run a marathon. A coach can help you with running form, provide workout schedules and provide feedback. But if you don’t run those workouts, you won’t build up the necessary speed and endurance for success.

The same principle applies for college mathematics classes; you really learn the material when you study it and do the homework exercises.

Here are some specific tips on how to study:

1. It is optimal if you can spend a few minutes scanning the text for the upcoming lesson. If you do this, you’ll be alert for the new concepts as they are presented and the concepts might sink in quicker.

2. There is some research that indicates:
a. It is better to have several shorter study sessions rather than one long one and
b. There is an optimal time delay between study sessions and the associated lecture.

Look at it this way: if you wait too long after the lesson to study it, you would have forgotten much of what was presented. If you study right away, then you really have, in essence, a longer class room session. It is probably best to hit the material right when the initial memory starts to fade; this time interval will vary from individual to individual. For more on this and for more on learning for long term recall, see this article.

3. Learn the basic derivative formulas inside and out; that is, know what the derivatives of functions like are on sight; you shouldn’t have to think about them. The same goes for the basic trig identities such as and

Why is this? The reason is that much of calculus (though not all!) boils down to pattern recognition.

For example, suppose you need to calculate:

If you don’t know your differentiation formulas, this problem is all but impossible. On the other hand, if you do know your differentiation formulas, then you’ll immediately recognize the and it’s derivative and you’ll see that this problem is really the very easy problem .

But this all starts with having “automatic” knowledge of the derivative formulas.

Note: this learning is something your professor or TA cannot do for you!

4. Be sure to do some study problems with your notes and your book closed. If you keep flipping to your notes and book to do the homework problems, you won’t be ready for the exams. You have to kick up the training wheels.
Try this; the difference will surprise you. There is also evidence that forcing yourself to recall the material FROM YOUR OWN BRAIN helps you learn the material! Give yourself frequent quizzes on what you are learning.

5. When reviewing for an exam, study the problems in mixed fashion. For example, get some note cards and write problems from the various sections on them (say, some from 3.1, some from 3.2, some from 3.3, and so on), mix the cards, then try the problems. If you just review section by section, you’ll go into each problem knowing what technique to use each time right from the start. Many times, half of the battle is knowing which technique to use with each problem; that is part of the course! Do the problems in mixed order.

If you find yourself whining complaining “I don’t know where to start” it means that you don’t know the material well enough. Remember that a trained monkey can repeat specific actions; you have to be a bit better than that!

6. Read the book, S L O W L Y, with pen and paper nearby. Make sure that you work through the examples in the text and that you understand the reasons for each step.

7. For the “more theoretical” topics, know some specific examples for specific theorems. Here is what I am talking about:

a. Intermediate value theorem: recall that if , then but there is no such that . Why does this not violate the intermediate value theorem?

b. Mean value theorem: note also that there is no such that . Why does this NOT violate the Mean Value Theorem?

c. Series: it is useful to know basic series such as those for . It is also good to know some basic examples such as the geometric series, the divergent harmonic series and the conditionally convergent series .

d. Limit definition of derivative: be able to work a few basic examples of the derivative via the limit definition: and know why the derivative of and do not exist at .

Having some “template” example can help you master a theoretical concept.

Part II: Attitude
Your attitude will be very important.

1. Remember that your effort will be essential! Again, you can’t learn to run a marathon without getting off of the couch and making your muscles sore. Learning mathematics involves some frustration and, yes, at times, some tedium. Learning is fun OVERALL but it isn’t always fun at all times. You will encounter discomfort and unpleasantness at times.

2. Remember that winners look for ways to succeed; losers and whiners look for excuses for failure. You can always find those who will be willing to enable your underachievement. Instead, seek out those who bring out your best.

3. Success is NOT guaranteed; that is what makes success rewarding! Think of how good you’ll feel about yourself if you mastered something that seemed impossible to master at first. And yes, anyone who has achieved anything that is remotely difficult has taken some lumps and bruises along the way. You will NOT be spared these.

Remember that if you duck the calculus challenge, you are, in essence, slamming many doors of opportunity shut right from the get-go.

4. On the other hand, remember that Calculus (the first two semesters anyway) is a Freshman level class; exceptional mathematical talent is not a prerequisite for success. True, calculus is easy for some but that isn’t the point. Most reasonably intelligent people can have success, if they are willing to put forth the proper effort in the proper manner.

Just think of how good it will feel to succeed in an area that isn’t your strong suit!

The potential harm in using outliers as examples…..

I think that this is common in this day and age: I have some students who are struggling in our “elementary conceptual calculus” course. They come to class, but work a large number of hours at a job in order to make ends meet. So…they are often left with very little time to study.

And yes, IN THIS COURSE, most of the students need to study quite a bit in order to have a chance at even a “C”.

In short: most students need to have a certain number of hours in order to sleep and to study..in addition to making the classes and their part time jobs.

Now, some might say that this is nonsense.

I remember a professor I had at the Naval Academy. He said that when he was an undergraduate he studied very little for his math classes as he paid his own way through school by waiting tables. He made up for it by PAYING ATTENTION IN CLASS.

That is well and good…..but then remember that he had an earned Ph.D. in mathematics from MIT.

Most of us don’t have that type of natural ability.

Yes, Mohammed Ali could break the conventional rules of boxing (dangle his arms, lean away from punches):

But most, including most other professional boxers, don’t have that kind of ability.

Yes, there are people who can run a 2:15 marathon on 35 miles a week of training:

Following the 1976 trials he trained by running 35 miles per week and ran “a 2:14:37 for second place at the Nike-Oregon Track Club Marathon in Eugene in 1978. After that, he ran 2:15:23 for 15th place in the Boston Marathon in 1979.”

But most of us aren’t that gifted (this was Tony Sandoval, cowinner of the 1980 US Olympic Trials Marathon)

Yes, some can make a successful film while being stoned on marijuana, but most of us aren’t as talented as the Beatles.

The list can go on and on. The bottom line: you can gain inspiration from the incredibly successful, but you won’t be able to get away with taking the short cuts that many of them got away with. Neither you nor I are outliers.

Rant: please stop with the teaching of “gimmicks for calculation”

I finished teaching calculus II (our course: techniques for integration, applications of integrals and infinite sequences/series) and noticed that some of our freshmen students came in knowing how to do many of the calculations…did well on the first exam…then didn’t do so well in the rest of the course.

Evidently, they were well versed in calculation tricks learned in high school; give them and they could whip out a table.

So here is my rant: we teach integration by parts not so much to calculate integrals like (which can be rapidly done with a calculator) but rather so they can understand the technique of integration by parts.

Why? Well, there are many uses of integration by parts and I’ll just display a few uses of them:

1. Taylor Polynomials. How do we get these? If we assume that has enough derivatives, we proceed in the following manner: calculate in two different ways: use the Fundamental Theorem of calculus on one side (to obtain and use integration by parts on the other side: (yes, we are being choosy about which anti derivative of to use).
This means: so and one proceeds from there.

2. Differential equations: given one seeks to find an integrating factor (which is so as to get:

which can be written as . That is, the left hand side is just the product rule for derivatives, which, as you know (if you are a calculus teacher), is really all integration by parts is!

Sure, one can jazz it up (as we subtly did in the Taylor Polynomial calculation); the integration by parts formula is really where are arbitrary constants. But, my main point is that integration by parts should be UNDERSTOOD; short cuts to do tedious calculations are relatively unimportant, IMHO.

Now if you want to ask students “why does tabular integration work”, then….GREAT!

What do you mean “you don’t know”???

I am grading calculus II exams and the above photo looks a bit like me. I’ll calm down before I hand the exams back to the students.

But I swear: I had one student DURING THE EXAM say “hey, you can’t do because and is undefined. Yes, this person passed calculus I and yes, we did that integral in calculus I (some schools hold off and develop using the Fundamental Theorem of Calculus). And yes, previously THIS SEMESTER we did integrals like .

(facepalm).