Skip to main content
30 votes
Accepted

Should an undergraduate math program contain a course on Lebesgue integration?

I think the existing answers understate how much a standard American math major does not see the Lebesgue integral. I'm going to poke around at a variety of college websites to see how they cover this ...
David E Speyer's user avatar
27 votes

How to properly define volume for beginner calculus students?

It depends somewhat on the style of the course, but the majority of calculus students do not need a formal definition of volume or area, in my experience. They have studied geometry and (usually) ...
user22788's user avatar
  • 854
17 votes

How to properly define volume for beginner calculus students?

$dV$ represents a tiny bit of $V$. $V = \int dV$ says that you can find the volume by adding up all the tiny bits of volume. This is why it is called an "integral;" you need to integrate all ...
Chris Cunningham's user avatar
15 votes

Should an undergraduate math program contain a course on Lebesgue integration?

Is it standard for a math undergraduate program to have a course on Lebesgue integration? No (assuming that "have a course" means "require people to take such a course in order to get ...
user19682's user avatar
  • 151
12 votes

A different symbol for the indefinite integral/antiderivative?

First, yes, many teachers use that term, "anti-derivative" and "integral" only when a definite integral is in use. For your examples of each, it seems to me that you offered a clear distinction ...
JTP - Apologise to Monica's user avatar
11 votes

Evaluating integrals geometrically, without using the fundamental theorem of calculus

A useful trick is the idea of rearranging values of a function. For example, while $\sin^2 \theta$ and $\cos^2 \theta$ have different graphs on $0 \leq \theta \leq \pi/2$ you can tell from the graphs ...
James S. Cook's user avatar
11 votes

Antiderivative of $1/x$, with or without absolute value?

Even $\int \frac{1}{x} \textrm{ d}x = \ln(|x|) + C$ is incorrect. It should be $$ \int \frac{1}{x} \textrm{ d}x = \begin{cases} \ln(x) + C_1 \textrm{ if $x > 0$}\\ \ln(-x) + C_2 \textrm{ if $x < ...
Steven Gubkin's user avatar
10 votes
Accepted

A very tricky pseudo-proof of $0=-1$ through series and integrals

[...]were you able to locate the fatal flaw at first sight? Given the context I was a priori quite certain it would be an issue in interchanging limits. What else could it be? You would not make an ...
quid's user avatar
  • 7,722
10 votes

What is the terminology for "self-referral" integrals in calculus?

To my knowledge, this is most commonly known as "cyclic" integration by parts.
Justin Skycak's user avatar
9 votes

Evaluating integrals geometrically, without using the fundamental theorem of calculus

In his Calculus book, Spivak gives these two exercises before he ever introduces the fundamental theorems of calculus. Evaluate without doing any computations: $$\int\limits_{-1}^{1} x^3\sqrt{1-x^2} \...
Mike Pierce's user avatar
  • 4,845
9 votes

How can I explain why numerical integration is easy, but symbolic integration is hard?

It might be fun to have your students pretend that the only functions they know are sums of monomials $cx^n$ where $n\in{\bf Z}$, and in particular, play like they know nothing about the function $f(x)...
user52817's user avatar
  • 11k
8 votes

How can I explain why numerical integration is easy, but symbolic integration is hard?

You should expect numerical integration to be "easier" [1] than symbolic integration because it is answering a fundamentally weaker question. That is, symbolic integration, if you can do it, gives you ...
Adam's user avatar
  • 5,873
8 votes

Intuition or geometry for Partial Fractions

Introduction I wasn't taught the partial fractions decomposition (PFD) in calculus. We didn't cover it in high school, and when I went to college, they assumed we all knew it. Somehow it was when I ...
user1815's user avatar
  • 5,760
8 votes

Why can an easily graphable definite integral, be labyrinthine to evaluate?

Calculating the definite integral is just as easy as graphing. You just plot your graph on a millimeter grid paper and count the number of squares below the graph. Of course, it does not give you an ...
Kostya_I's user avatar
  • 1,411
8 votes

Should an undergraduate math program contain a course on Lebesgue integration?

Is it standard for a math undergraduate program to have a course on Lebesgue integration? Yes, and I find it bizarre that a university would not have one. Lebesgue integration (or measure theory more ...
otah007's user avatar
  • 189
7 votes

Integral calculus from the modern viewpoint

There are many techniques of integration. Some of them, like integration by parts, are important theoretically. Integration by parts shows up in the derivation of the Euler-Lagrange equations in ...
Adam's user avatar
  • 5,873
7 votes

Integral calculus from the modern viewpoint

1: Up until the collegiate level, most math students memorize the 'right' way to do certain types of problems, and then repeat that hundreds of times on homework, so that they can repeat it on a test. ...
charmoniumQ's user avatar
7 votes

Double Integral: Area or Volume?

You are integrating a function $z=f(x,y)$ but the units of $z$ do not have to be for length. The units could be for mass density, in which case the units of the double integral would be for mass. Or $...
user52817's user avatar
  • 11k
7 votes

A different symbol for the indefinite integral/antiderivative?

I'll echo other responses with the same: Do NOT introduce made up notation. I've made the effort in my Calculus courses to follow your outline while avoiding any new symbols. Similar to user20311's ...
Aeryk's user avatar
  • 8,019
6 votes
Accepted

Double Integral: Area or Volume?

A double integral represents integrating over an area. If the integrand is a height (i.e. with units of length), the result will be a volume (i.e. units of length$^3$). If the integrand is a flux (e.g....
PGnome's user avatar
  • 296
6 votes

Evaluating integrals geometrically, without using the fundamental theorem of calculus

The integrals of sine and cosine have a nice geometric interpretation, if you and your students are comfortable treating the differential of a circular arc as equivalent to a straight hypotenuse (in ...
user1815's user avatar
  • 5,760
6 votes

What is a good way to explain the Lebesgue integral to non-math majors?

I have the impression that the underlying problem is the expected value itself, not the integral (on which the expected value is based, of course). But since the question asks about the integral, I ...
Michał Miśkiewicz's user avatar
6 votes

A different symbol for the indefinite integral/antiderivative?

Before inventing new notation, it is very important to learn the accepted notation and teach it to your students correctly. The question contains the following claim: $$\int \cos = \sin$$ which is ...
Chris Cunningham's user avatar
6 votes

Should an undergraduate math program contain a course on Lebesgue integration?

Yes. As an elective. If there are interested students we ought to give them a platform to exercise their curiosity. As a requirement, well, that depends on what you aim for the mission of your program....
James S. Cook's user avatar
6 votes

Should an undergraduate math program contain a course on Lebesgue integration?

For undergraduate maths students at the University of Oxford, Riemann Integration is mandatory for 'freshmen' (year 1), and Lebesgue Integration is an elective taken by 'sophmores' (year 2). Full ...
J Marza's user avatar
  • 161
6 votes
Accepted

Student forgets to remove dx after integrating

Since I teach physics, where units are very important, I would suggest considering $x$ to be a length (carrying units of meters). The left-hand-side interpreted like an area has units of $m^2$. The $x^...
robphy's user avatar
  • 592
6 votes
Accepted

Definite integrals with equal limits

How would explain this to students? Explain that checking that conditions are satisfied is part of the process of applying theorems/laws. (Admittedly, though, since integration change of variable is ...
ryang's user avatar
  • 1,832
5 votes

Tables of primitives with indication of solution method

I don't think there exists a published reference book that goes into as much detail as you want with the kind of very elementary integral you gave as an example (a simple exponential times a ...
Dave L Renfro's user avatar

Only top scored, non community-wiki answers of a minimum length are eligible