# Tag Info

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 ...
• 5,332

### 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

### 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 ...
• 21.7k

### 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

### 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 ...

### 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 ...
• 10.9k

• 4,845

• 11k

### 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 ...
• 8,019
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....
• 296

### 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 ...
• 5,760

### 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 ...

### 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 ...
• 21.7k

### 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....
• 10.9k

### 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 ...
• 161
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^...
• 592
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 ...
• 1,832