# Tag Info

### 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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### 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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### 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,...
• 4,314
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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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### 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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### 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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### 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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### 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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### 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 ...

### 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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### 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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