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 ...
- 5,243
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) ...
- 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 ...
- 21.2k
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 ...
- 151
14
votes
Should we teach trigonometric substitution?
In reality, I think this is not the most important topic, and if I was designing a curriculum from scratch, I would probably omit it.
We rarely have that luxury however. In my state, for example, ...
- 24.3k
12
votes
Accepted
Demonstrating that integrals of some unbounded functions exist, and others do not
When we write $\int_0^1 \frac{1}{\sqrt{x}} d x = 2$, we do not mean that the Riemann integral of $\frac{1}{\sqrt{x}}$ on $[0,1]$ exists and is equal to $2$, because, as you note, this is false. One ...
- 24.3k
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 ...
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 ...
- 10.7k
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♦
- 7,652
10
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
$$
\int \frac{1}{x} \textrm{ d}x = \begin{cases}
\ln(x) + C_1 \textrm{ if $x > 0$}\\
\ln(-x) + C_2 \textrm{ if $x < ...
- 24.3k
9
votes
Evaluating integrals geometrically, without using the fundamental theorem of calculus
In his Calculus book, Spivak gives these two exercises before he ever introduces the fundamental theorems of calculus.
Evaluate without doing any computations:
$$\int\limits_{-1}^{1} x^3\sqrt{1-x^2} \...
- 4,606
9
votes
Should we teach trigonometric substitution?
Yes. We should teach trigonometric substitution. But, I take it a step further, I think we should also teach hyperbolic substitution. With this additional technique the idea of the substitution is ...
- 10.7k
9
votes
Should we teach trigonometric substitution?
My first feeling on this is that, although in any given calculus course there might be a reason not to introduce trig substitution, in practice it is still a valuable technique that takes a whole huge ...
- 5,970
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)...
- 10.3k
8
votes
What is a good way to explain the Lebesgue integral to non-math majors?
To see the reason why Lebesgue integral is preferred in probability theory one must go beyond the setting of real functions $f \colon \mathbb{R} \mapsto \mathbb{R}$. In this setting both the Riemann ...
- 1,758
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 ...
- 5,192
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 ...
- 5,345
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 ...
- 1,164
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 ...
- 189
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 $...
- 10.3k
7
votes
Integral calculus from the modern viewpoint
1: Up until the collegiate level, most math students memorize the 'right' way to do certain types of problems, and then repeat that hundreds of times on homework, so that they can repeat it on a test.
...
- 191
7
votes
Integral calculus from the modern viewpoint
There are many techniques of integration. Some of them, like integration by parts, are important theoretically. Integration by parts shows up in the derivation of the Euler-Lagrange equations in ...
- 5,192
7
votes
A different symbol for the indefinite integral/antiderivative?
I'll echo other responses with the same: Do NOT introduce made up notation.
I've made the effort in my Calculus courses to follow your outline while avoiding any new symbols. Similar to user20311's ...
- 7,500
6
votes
What is a good way to explain the Lebesgue integral to non-math majors?
I have the impression that the underlying problem is the expected value itself, not the integral (on which the expected value is based, of course). But since the question asks about the integral, I ...
6
votes
Accepted
Double Integral: Area or Volume?
A double integral represents integrating over an area. If the integrand is a height (i.e. with units of length), the result will be a volume (i.e. units of length$^3$). If the integrand is a flux (e.g....
- 276
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 ...
- 5,345
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