45

On quizzes, homeworks, and tests, I repeatedly ask questions like this: Find three different functions that have derivative equal to $x^2 + x$. Forcing them to do antiderivatives and deal with the quantifier on the +C without staring at the notation helps some of them separate the +C from the voodoo magic. I do a similar thing in college algebra classes ...


27

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 variables. These have somewhat confusing, perhaps even counter-intuitive names. So, I will use "local" as synonymous with "bound" and "non-local" as synonymous ...


24

I go a step further than Thomas (see Henry Towsner's answer). In my view, $$ \int f(x) \ dx = \{ F(x) \ | \ F'(x)=f(x) \} $$ On a connected domain, it is true that $F'(x)=G'(x)$ implies $F(x)-G(x)=c$ hence, given an integrand which is continuous (or piecewise continuous, insert your favorite weakened set of functions here) we may write: $ \int f(x) \ dx = \{ ...


15

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, I add, but we don't integrate for the sake of integration. For us, it is a practice of problem solving in which we need to choose the right techniques from ...


14

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, all state colleges must meet certain common standards to be able to transfer credit: one of these standards is teaching trig sub. If I had to include it, I ...


13

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 variable' which I've described in the past as pedagogical cancer. There's no such thing as the function $f(x)$ (unless $f$ is a functional, but that's a different ...


12

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 $[0,a]$ and label it $f(x)$. Now we want to figure out what $g(x)=f(a-x)$ looks like. By direct computation, $g(0)=f(a)$ and $g(a)=f(0)$. Once you have ...


12

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 partition, rather than the "heights" of the rectangles, can be used to encode the "shape" of function being integrated. The way to encode the function in the ...


11

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.com, which shows how things can break when switching to multiple variables. It provides the following basic example of a possible error when handling partial ...


11

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: $$\int_{\phi([a,b]}f(x)\,dx=\int_{[a,b]} f(\phi(t))|\phi'(t)|\, dt$$ In the second, you do need $\phi$ to be monotone. In the first formulation, you do not need this assumption. Of course when you apply the ...


11

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 can construct sequences of riemann sums for this function which converge to any number $>2$ than you wish, or sequences which diverge to infinity, or ...


11

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 of $\sin \theta$ and $\cos \theta$ that the values which the squared functions take must be equal. In short, $$ \int_{0}^{\pi/2} \cos^2 \theta \, d\theta = \...


11

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 between the two expressions, i.e. that the bounds is what what makes the integral definite. For me, that's where the explanation to students tends to reach a ...


10

[...]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 index error or something like this. Yet, this is a bit 'cheating' though. It is a nice problem. Do you think it is a well-suited exercise for Calculus-2 (...


9

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 class of integrals and "make sense" of them. For instance, to tell whether you have just fed your CAS junk - misplace an $x^2$ by and $x^3$ and it's all over! ...


9

In his Calculus book, Spivak gives these two exercises before he ever introduces the fundamental theorems of calculus. Evaluate without doing any computations: $$\int\limits_{-1}^{1} x^3\sqrt{1-x^2} \,\mathrm{d}x \qquad\qquad\int\limits_{-1}^{1} \left(x^5+3\right)\sqrt{1-x^2} \,\mathrm{d}x$$


9

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 $f(x)=\ln(x)$. You see where this is going: we can easily graph $f(x)=x^n$ between $x=1$ and $x=2$ for any $n$, and find an antiderivative with the one annoying ...


9

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 exact answer, only an approximation, but so is graphing. If it is easy to discern some approximate or qualitative features of the problem, it does not mean that ...


8

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 integral. That said, when I was a graduate student, I came across this book Lebesgue Integration and Measure by Alan J. Weir that, instead of connecting the ...


8

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 much clearer. Also, teach it with confidence. Teach it as if they can all understand it... because they can. Why do students have trouble with this topic? I ...


8

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 an infinitely precise answer. Numerical integration gives you an answer which is approximate, ideally with some bound on the amount of error. It is like the ...


8

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 read the proof in van der Waerden's Modern Algebra that I understood why it was called partial fractions. I looked at it again a couple days ago, and ...


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