27 votes

How to convince students of the integral identity $\int_0^af(x)dx=\int_0^af(a-x)dx$?

Problem of sloppy notation The notation is sloppy. Your students are justifiably confused. We've just gotten used to it. In order to untangle this, we need the notion of free variables and bound ...
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  • 421
23 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 ...
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15 votes

How to give homework for integration techniques?

First of all, I try to be honest with my students by telling them directly and explicitly about the existence of such integration machines (It is silly of me assuming that they don't know that!). Then,...
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  • 4,314
14 votes
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What is a good way to explain the Lebesgue integral to non-math majors?

As you told the student, the easiest way is to regard the Lebesgue integral as beginning with a partition of the range, rather than the domain. Perhaps a more refined way to view this is that the ...
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  • 5,877
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, ...
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14 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 ...
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  • 141
13 votes

How to convince students of the integral identity $\int_0^af(x)dx=\int_0^af(a-x)dx$?

This answer attacks not only this problem, but a lot of all others. At the expense of going against the grain, however. A much deeper issue is this permanent grip on the concept of 'function of a ...
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  • 824
12 votes

How to convince students of the integral identity $\int_0^af(x)dx=\int_0^af(a-x)dx$?

I would ask students to consider what the graphs of $f(x)$ and $f(a-x)$ look like (and how they are related to each other) on the interval $[0,a]$. Draw a sketch of some arbitrary-looking function on ...
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  • 16.4k
11 votes
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Should $\varphi$ be monotone in the integration by substitution?

There are two formulations for definite integrals: $$\int_{\phi(a)}^{\phi(b)} f(x)\, dx=\int_a^b f(\phi(t))\phi'(t)\, dt$$ and the one you state: $$\int_{\phi([a,b]}f(x)\,dx=\int_{[a,b]} f(\phi(t))...
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  • 7,703
11 votes
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Non-Rigorous Use of Differentials

As @BenCrowell mentioned, the transfer principle proves that direct algebraic manipulation of infinitesimals in single variable calculus is allowed. But I stumbled upon this post in theshapeofmath....
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  • 684
11 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 ...
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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 ...
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11 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 ...
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10 votes
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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 ...
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  • 7,592
9 votes

Introducing the Lebesgue integral before Riemann's

In the US, you'd be hard-pressed to find any student seeing the Lebesgue integral before having ever seen the Riemann integral. Every calculus book I've seen defines the integral as the Riemann ...
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  • 2,936
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 ...
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  • 5,760
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} \...
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  • 4,480
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)...
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  • 7,703
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 ...
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8 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 ...
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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 ...
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  • 4,784
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 ...
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  • 4,619
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 ...
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  • 689
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 ...
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  • 189

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