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Questions tagged [integration]

For questions related to the teaching of integral calculus

5
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1answer
112 views

Using discrete examples in the beginning of integration

In Germany, one usual example to start teaching about integrals is to look at a simple (piecewise constant or with constant slope) functions that make up a water flow vs. time diagram and ask about ...
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2answers
330 views

Why do we state the antiderivative of $\sec x$ as $\ln |\sec x +\tan x|$?

One easy integration of $\sec x$ substitutes $u=\sin x$, viz.$$\int\frac{\cos x}{1-\sin^2 x}\,\mathrm{d}x=\frac{1}{2}\ln\left|\frac{1+\sin x}{1-\sin x}\right|+C.$$Multiplying top and bottom by $1+\sin ...
4
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2answers
239 views

Would it be constructive to teach a whole course on how to evaluate certain hard integrals?

For example, there is an integration bee and lots of youtube videos tackling hard or impossible integrals with just tools of Calculus I & II. However, a lot of these integrals found on youtube ...
9
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3answers
333 views

Integral calculus from the modern viewpoint

This is a soft question. What is the purpose of teaching techniques of integration at the college level? More specifically, in the sense of putting integration into practice, what value does ...
5
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1answer
94 views

Tables of primitives with indication of solution method

I am looking for an extensive source (often called "table of integrals") listing primitives of various classes of functions including the "elementary" ones (rational functions, functions involving ...
8
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1answer
194 views

Integration by substitution not using letter u

Until a few days ago, every calculus textbook I have seen uses $u$ as the default variable for integration by substitution (a.k.a. integration by change of variables). It was brought to my attention ...
6
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2answers
126 views

Alternative ways of thinking about the one-variable Riemann integral for elementary calculus,

I think I've done a decent job with teaching my students limits and derivatives so far in elementary calculus -- they were particularly intrigued with how easy and how accurate a first-order, linear ...
3
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0answers
202 views

How do i deal with students who make these mistakes? [closed]

I came across some interesting mistakes in many area of mathematics with my students and do not let me also to tell you for university students level, I would like to know How do i deal with ...
16
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9answers
697 views

Evaluating integrals geometrically, without using the fundamental theorem of calculus

I'm designing a lesson for an Introduction to Integral Calculus class, and I want to encourage students to evaluate integrals without just going straight for the antiderivative and using the ...
0
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4answers
286 views

Double Integral: Area or Volume? [closed]

When we study double integral many Calculus textbooks state that for a region $R$ in the plane $$\iint_R1\ dA= \text{area bounded by }R $$ But double integral actually give the volume of a solid. ...
10
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1answer
425 views

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

Dealing with a recent question I spotted a very nice exercise for Calc-2 students, i.e. to find the mistake in the following lines. Lemma 1. For any $n\in\mathbb{N}$, we have: $$ \int_{0}^{1} x^n\...
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2answers
234 views

Demonstrating that integrals of some unbounded functions exist, and others do not

This is my first year teaching calculus. On a recent quiz, I asked my students to give an argument that $\int^0_1(1/x)dx$ does not exist. I was looking for arguments that appealed to Riemann sum ...
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8answers
1k views

Should we teach trigonometric substitution?

This is the question that was not asked here. Also related is this question, but both presuppose that it will be taught and ask about how best to do it. My question here is, suppose we are designing ...
9
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1answer
456 views

Should $\varphi$ be monotone in the integration by substitution?

I'm trying to calculate $$\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}\sin t \cos^3 t\,dt$$ using integration by substitution $$\int_{\varphi([a;b])} f(x)dx=\int_{[a;b]} f\left(\varphi(t)\right)|\varphi'(t)|...
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5answers
2k views

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

A common identity in integration is $\int_0^af(x)dx=\int_0^af(a-x)dx$. The steps to prove it (algebraically, ignoring the geometric method) are as follows. Let $u=a-x$ so $dx=-du$. $\int_0^af(a-x)...
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7answers
555 views

How to give homework for integration techniques?

When I was a freshman in Mathematics we learned the usual integration techniques (lots of standard integrals, integration by parts, substitution, partial fractions,…). As homework we simple got a ...
18
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7answers
583 views

Non-Rigorous Use of Differentials

Consider the following example of working "directly" with differentials. One way to approach the problem of determining the arc length of the graph of a single-variable function is to imagine the arc ...
11
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1answer
504 views

Language to Distinguish Between Variables and Arbitrary Constants

Today in second semester calculus, I found myself stumbling a bit to provide a natural-sounding explanation for all the letters involved in the expression $$ \lim_{t \rightarrow \infty} \int_1^t \frac{...
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2answers
2k views

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

A few days ago I had my last discussion session on probability theory as a TA. In the end I asked students to ask me questions as this is the last class. One of the student asked me about the (real) ...
3
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0answers
97 views

How useful/useless is the indefinite integral [duplicate]

After having met yet another person confused by indefinite integrals today, I've finally decided to ask the community. Do you think it makes sense to teach indefinite integrals? My opinion is that ...
13
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2answers
735 views

Introducing the Lebesgue integral before Riemann's

Has anyone attempted to introduce, or has data on such endeavor, Lebesgue integration before Riemann? I've seen many discussions about how the Riemann integral is obsolete and that it is presented ...
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9answers
3k views

Should we avoid indefinite integrals?

I am very uncomfortable with indefinite integrals, as I have a hard time giving them a precise sense that matches the way they are written and the usual meaning of other symbols. For example, when ...