Questions tagged [integration]

For questions related to the teaching of integral calculus

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47 votes
10 answers
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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 ...
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24 votes
6 answers
3k 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) ...
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22 votes
8 answers
2k 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 ...
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21 votes
7 answers
840 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 ...
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21 votes
8 answers
840 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 ...
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17 votes
5 answers
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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  • 1,076
16 votes
9 answers
1k 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 ...
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  • 4,480
15 votes
2 answers
607 views

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

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 ...
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  • 499
15 votes
2 answers
1k 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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  • 2,879
12 votes
1 answer
628 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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11 votes
3 answers
389 views

Usefulness of $u$-substitution in and beyond early Calculus?

My students, when presented with an integral (source) like $$\int (2x+2)e^{x^2+2x+3} \ dx$$ are apt to recognize derivative patterns like $u' e^{u}$ and reverse-engineer anti-derivatives rather than ...
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  • 788
11 votes
1 answer
629 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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10 votes
5 answers
1k views

A different symbol for the indefinite integral/antiderivative?

Examples. An indefinite integral (or antiderivative) of $\cos$ is $\sin$: $$\int \cos = \sin.$$ Edit: There has been much unexpected confusion with the above statement. I define the above statement ...
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9 votes
3 answers
396 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 ...
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9 votes
1 answer
1k 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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8 votes
5 answers
3k views

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? Does Riemann integral suffice for undergraduates? The reason of my question is I read a paper by Bartle titled ...
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  • 183
8 votes
1 answer
582 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 ...
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  • 1,988
7 votes
7 answers
1k views

How can I explain why numerical integration is easy, but symbolic integration is hard?

I'm asking about definite integrals that can effortlessly be found numerically by high schoolers using software. For example, $$\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\...
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6 votes
1 answer
188 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 ...
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  • 161
6 votes
2 answers
165 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 ...
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5 votes
2 answers
616 views

Intuition or geometry for Partial Fractions

When teaching partial fractions, there's probably no way to escape the heavy algebra necessary for partial fractions, but I'm wondering how to introduce the idea in a way that is intuitive or ...
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4 votes
2 answers
521 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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4 votes
2 answers
279 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 ...
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  • 797
4 votes
2 answers
472 views

The hardest case of integration by partial fractions

The context is explaining to calculus students how to integrate rational functions by using partial fractions decomposition. As we all know, partial fractions decomposition is a method to write every ...
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  • 569
4 votes
1 answer
130 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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  • 2,191
4 votes
2 answers
161 views

Good Examples of Equations Derived from Elementary Calculus

I'm collecting additional enrichment content for my calculus students. I'm looking for examples of equations that are used in various fields, but which can be derived at least somewhat ...
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  • 919
3 votes
6 answers
2k views

Why can an easily graphable definite integral, be labyrinthine to evaluate?

How can I explain to 16-year-olds, who just started calculus, why it's so nettlesome and tricky to symbolically integrate definite integrals, when their graphs look so unremarkable and straightforward?...
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3 votes
4 answers
390 views

Student forgets to remove dx after integrating

I am tutoring another US college student in a Calculus 1 class. Initially, she was having trouble with basic concepts, but after much prodding most of the conceptual difficulties seem to have been ...
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  • 137
3 votes
1 answer
118 views

The purpose of a particular rational function integration exercise

This might be a more appropriate question for math.stackexchange, but it's about a problem I'm considering giving my students, so here it goes. One of the later exercises in Section 7.4 of James ...
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  • 131
3 votes
0 answers
221 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 ...
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3 votes
0 answers
132 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 ...
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  • 131
2 votes
2 answers
134 views

Analogy for cylindrical shells

The analogy for cross-sections is easy since we can think of how slices of bread can make up a loaf. But what would be the analogy for cylindrical shells? Regarding shapes, apparently there's ...
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1 vote
2 answers
100 views

Retain problems and combat regression in learning

Regressive Learning It's a really stressful situation. I can achieve but not retain expertise in maths problems. History 6 months back, I studied integration in Calculus at college. I learnt it all ...
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0 votes
4 answers
418 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. ...
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-2 votes
1 answer
113 views

Finding an error in a partial integration [closed]

There must be an error in this partial integration but I do not see it. Do you see it?
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