EDIT: I finally understood calculus after taking introduction to real analysis, and it was amazing because for the first time all the hand-waving disappeared and could be replaced with rock-solid arguments and increasing levels of abstraction (starting from the very definition of what the real numbers are). This is also important because functions can get very pathological[0][1][2]
[0] https://en.wikipedia.org/wiki/Weierstrass_function (continuous everywhere but differentiable nowhere)
[1] https://en.wikipedia.org/wiki/Cantor_function (derivative is zero almost everywhere but f(x) goes from 0 to 1)
[2] https://en.wikipedia.org/wiki/Thomae's_function (continuous at irrationals but discontinuous on rationals)
But those weed-out classes test a bunch of arcane mechanics and memorized formulas and transformations in the most intentionally obtuse exam questions possible. If you don't know the "one weird trick" you're kind of screwed.
Math weed-out classes are a lot like the tech interview problem, but for STEM majors.
Calc II was a class where it just wasn't possible because there was too much to derive on any given exam. It took me 1/2 way through the course to course-correct and make flashcards and such nonsense. Unfortunately that was the last (non-discrete) math class I took, so I never discovered what happens next.
To help my kids stay interested in math, I offered them the following promise: "Suspend your judgement until you get a chance to do proofs, because proofs are when math comes alive." One of my kids became a math major.
ug calculus was about having algebraic/trig manipulations memorized along with a table of transforms and a handful of tricks; where the actually beautiful ideas that if we use an infinitely "elastic" representation of numbers, we can solve hard approximation problems both correctly and easily- get totally glossed over.
physics has the same problem. basic physics without calculus is just a bunch of rote memorization. the idea that such a thing is taught and that is somehow "easier" is nuts. they should be taught together, as they were developed, as many of the expressions given to undergrads in physics are simply definitions of integrals and derivatives applied a few times.
no student in calculus should ever be wondering what the constant is for in a computed integral and no student in physics should wonder where the constants come from in equations of motion. (there should be no equations of motion, just definitions in integral/derivative form and definitions of integrals and derivatives)
Solution: spend more time on epsilon-delta so that students have time to wrap their minds around the idea. But I think the engineering departments would complain that the students who we send on to them cannot do basic computations. Also students would complain that we spend too much time on theory and not enough on application. There are probably other reasons that someone more experienced would know about.
On the other hand, I really enjoyed and did well in classes like real analysis, functional analysis and measure theory, even though I am not a mathematician.
But those classes are fun. Say, the construction of measures and Lebesgue type integrals makes sense to me.
Integration by parts? And this stuff? I feel like a stupid monkey. It's weird.
For a real mathematician, both of these types of skills are probably important. I wish I could do integrals better.
The only concept from that time I used in my day job is the binomial coefficients. Yet I don't regret taking the class, in some inexplicable way I feel it has made be better (at what I have no idea).
The only thing I really "learned" from studying calculus that I actually apply to the real-world is the ability to slow down and take each part individually. When I first took calculus, I tried to rush through the problems and inevitably dropped something important. It wasn't until I started forcing myself to write down each step (even if I thought I could do it in my head) that I started actually getting the right answers, and knowing that I actually had.
For actual math requirements, you started with real and complex analysis, abstract algebra, geometry and topology, and then some applied math classes such as partial differential equations, or numerical methods. There was also a requirement for probability and statistics.
One way you can tell the difference between a general requirement class and a major class is the size of the classroom and the majors taking the course. If you are in an auditorium with 300 freshmen taught by a TA and almost no one else in that course is a math major, then you are looking at a university requirement rather than a college requirement.
University requirements are not intended to weed anyone out, that would be contrary to the goals of the university. They should be doable by all who are admitted. When I went to grad school, I had the pleasure of teaching some of these calculus classes, and no one considered this to be a weed out class or a math major class. All the math majors we had tested out of calculus in high school, and most of our students had humanities majors (as the STEM students also tended to test out of it). Giving those humanity majors lots of tricky problems in order to try to weed them out from their own majors wouldn't make any sense.
Moreover the key skill in being a math major is the ability to do proofs. So the weed out classes tend to be real analysis or abstract algebra, as these are the classes where students first do proofs. As there are traditionally no proofs in calculus classes (the books may provide proofs, but you are not tested in your ability to prove theorems, but in your ability to calculate). Thus it wouldn't be a good weed out class for math majors even if it wasn't a general requirement class taken by all majors.
Real analysis is the version of these things taught to math students, rather than the (often mostly service) version that is taught for other programs (engineering, physics, etc.) that need calculus. It is unfortunate that many programs are structured so you can't even see this before surviving the standard 1st year calc progression, especially at large universities.
How math-oriented or not a particular program is varies obviously, but it's pretty common to see this distinction. When I was an undergraduate, entry to the honors math program ignored all calculus classes and results entirely (if I recall correctly, it was based on having a 1st class standing in linear and modern algebra courses). I think was entirely possible to complete a math major with little or no calculus at all.
I started freshman year intending to major in math, started with Calc 3. When the average grade among my classmates on the first test was a 56/100, curved of course, I knew something had to give. This was not the fun math I knew from before. A+ student up until this moment.
some kind of awesome integrated class where kids work with robot toy cars comes to mind as an interesting way it could be presented. (start by measuring their behavior and collecting data, computing crude integrals on the computer, moving into analysis using the reals)
I wonder what that was. (In my world it was just a bunch of axioms.)
Lots of schools break up calculus in different ways, and that's fine. My school (and the schools of lots of people I know) break calculus up into calc 1, which is limits, differentials, and a toe dipping into integrals. Calc 2 is the 8 or so different tools for integrating progressively more difficult integrals. Calc 3 is multivariate calculus.
Lots of people have a very difficult time with calc 2. It feels very plodding- calc 1 and calc 3 (and diff eq and lin alg...) felt like I was learning new insight every week, calc 2 just felt like memorizing new vocabulary words. It wasn't just that it was hard, it was that it was hard and boring.
(obviously if a school breaks calc up differently your experiences will probably be different)
It's good to understand things conceptually. But once I got the concept of integrals as the area under a curve, it felt like a lot of grunt work to learn so many tactics for solving them. But most of my focus has been on computers rather than pure mathematics. For pure math, it probably makes more sense to learn as many different methods as possible.
* While computational tools will symbolically solve a lot of integrals, they won't solve them all. Resorting to numerics often means you have lose some understanding along the way, because you no longer have a closed form expression to analyze.
* One general strategy in Physics is to take a complicated expression and make different sets of simplifying assumptions to reduce it to simpler forms. This adds explanation to your model because you understand how the system is said to behave under different limitating scenarios. But if you are not adept at manipulating complicated expressions, you won't be able to use the strategy fully. Computer solvers are really bad at writing mathematical expressions in the nicest way possible so that the simplifying assumptions pop out naturally.
Full disclosure: I am a physicist, who uses Mathematica quite a lot to solve various expressions (but I know the limitations of the tool).
The one I can think of offhand is the pendulum problem, where sin(theta) is approximately equal to theta for small values. But you made it sound like there are problems with multiple parts, and many different simplifications.
A dangerous trap IMO- both are valuable but incredibly different. The former is teaching a narrow-scoped trade as opposed to learning a full-fledged engineering discipline. The latter is much more generalizable and equips you to understand / build / use most tools going forward, rather than overfitting use only to the current fad of high-level tooling.
Bayesian statistics has been liberated by the ability to perform many dimensional integrations on the kind of likelihood functions appropriate for the problem, where before the advent of modern computational techniques Bayesian statistics had a reputation for concentrating on artificial problems that we happened to know how to solve.
Rewriting formulas in different forms can allow you to see analogies between them, allowing you to prove "mini-theorems", which you can use to make computations more efficient or to adapt slightly different mathematical tools to your problem.
Those things happen frequently even if you are a "just" a developer, (not necessarily with integrals/real analysis, for example combinatorics tricks are extremely common), but it's definitely a nice tool to have.
I took a single master's course as a deal to get my bachelor's and that was Random Signals and Stochastic Processes. Wow, you cannot get these concepts without a super strong mathematics background.
To this date I think it was both the hardest and most fulfilling course I took.
I have been on both sides of that, and far prefer to know what I'm doing.
Computer Algebra is actually not great at solving integrals (and the problem is unsolvable in general). But it’s not extremely common that one needs to symbolically integrate gnarly expressions and you can look up the tricks when it comes up.
Much in the same way that I believe introductory linear algebra is bogged down by endless matrix computation without a computer, I think forcing students to compute a million different gross integrals quickly has diminishing returns.
That is a useful trick to keep in mind all the time. The point of course, is not to learn to solve integrals, but to learn a transferrable piece off mental jiujitsu.
Even being able to do simple arithmetic in your head to check for errors in slides/talks is a major step up. It will only take one moment of realising that the person talking has made a mistake in their calculation, and is basing their argument on that mistake, to realise the power of this - and that's just the simple stuff.
However, when I'm working on something that heavily uses a specific mathematical method I like to dig in and deeply understand that aspect, otherwise you can become dependent on other peoples implementations that may not be optimal for your use case.
Like, https://www.wolframalpha.com/input/?i=integrate+u%5En+sqrt%2...
I looked at that and realize that I'd have no future as a physicist and switched to CS.
I was too naive to do so, but if anyone out there is in a class and the official suggested book doesn’t help you, ask the Internet for a respected alternative.
There's an algebra of differentials that was formalized quite late (I think at the 19th century) but accepts all of the operations you can use for scalars. The chain rule is just fraction simplification.
https://courses.cs.washington.edu/courses/cse446/18wi/slides...
Not my best professor.
Getting the answers before the test was the alleged cheating.
1: That one time was not even in school; when I graduated I applied to OCS, and the recruiter told me point-blank that his superiors thought I cheated on the test. It was apparently the highest score anyone at this particular recruiting site had ever seen, and my grades were mediocre (see above about poor study habits).
Did you not put the steps on the paper you submitted?
But to be fair, as a freshman in a college calculus class that was essentially reviewing what I had already done in high school I was kind of an asshole. I skipped a lot of his classes and often did the formulaic questions in my head and just wrote down the answer.
I don't recommend this approach.
int_0^pi ln[1 - cos(x)] dx = - pi ln[2]
Is this easy?
One observation is that by symmetry
int_0^pi ln[1 - cos(x)] dx = int 0^pi ln[1 + cos(x)] dx
and so the calculation is equivalent to
1/2 int_0^pi ln[ sin(x)^2 ] dx = 2 int_0^pi/2 ln[ sin(x) ] dx.
Now we can use a similar trick again:
int_0^pi/2 ln[ sin(x) ] dx = int_0^pi/2 ln[ cos(x) ] dx
so
4 int_0^pi/2 ln[ sin(x) ] dx = 2 int_0^pi/2 ln[ sin(x) cos(x) ]
= 2 int_0^pi/2 ln[1/2] dx + 2 int_0^pi/2 ln[sin(2x)] dx
= - pi ln[2] + int_0^pi ln[sin(u)] du
(using substitution u = 2x)
= -pi ln[2] + 2 int_0^pi/2 ln[ sin(x) ] dx
and so the original integral is -pi ln[2].
1 - cos(x) can be written as 2*cos^2(x/2)
Take the logarithm and you get 2ln (cos(x/2)), which is relatively common and the solution is https://www.quora.com/How-do-I-integrate-log-cos-x-from-0-to...
It's much easier to prove that the difference between the integral of log(a^2 - 2 a cos(x) + 1) and 2 pi log(a) goes to 0 when a goes to infinity.
Incidentally, your correct observation is in contradiction with f(a) = 2 pi log(|a|) since the latter only holds for |a| >= 1. This is because f(a) is not a smooth function.
In high school, I transformed x^x into e^(x log x), expanded that into the Taylor series, and integrated term by term. I got a "solution", but it wasn't closed form - it was an infinite series. And, for a given error limit, it probably converged more slowly than a decent numerical integration. So, not worth much. But I "solved" it...
Fifteen years ago and still terrifies me.
One question: It mentions that Wolfram alpha will fail on integrals that this trick can work for. Is that just because it will time out (we need more compute) or is the trick difficult to automate?
Differentiation under the integral sign only works for certain well behaved functions and isn't easy to automate since you now need to figure out where to parametrize and you don't have good structure theorems to help you.
IMO contour integration is a more powerful and easier to intuit technique.
I mean, sure, we have simpler rules for dfferentiation, but _why_?
I sometimes wonder if it's differentiation is P and integration is NP (for the restricted case of functions where the primitives are "nice")
The most straightforward definition of indefinite integration on the other hand is “the inverse of differentiation”. Like a lot of functions, inverses are much more complicated. Even in the case of matrix-vector multiplication, the inverse function (the inverse matrix) has an expression in terms of determinants in the original entries of the matrix - much more complicated than the original problem.
So I think that it’s the more simple theme of: even if the function is easy to describe, its inverse may be extremely complicated.
The more abstract reason differentiation is easier than integration is that because differentiation is a local operation, given a family of functions differentiation will be closed within that family. That is the derivative of an elementary function is an elementary function, the derivative of a rational function is a rational function, the derivative of a function f is a function f' that is within the same family of f and will share almost all of the same properties as f.
Integration on the other hand does not have this property, and in fact it's very difficult to find family of functions that are closed under integration. The integral of a rational function may not be rational, the integral of an elementary function may not be elementary. Integration is a global operation whose solution may require the construction of a new family of functions whose properties are entirely unlike the function being integrated.
It's like how squaring a number is usually closed within that number system, but taking the square root of a number could require creating entirely new number systems, whether it's real numbers or complex numbers (at which point the square root is closed). With integration, the issue is that it never ends up being closed, you can use integration on a family of functions to jump out of that family and construct entirely new families of functions seemingly without end.
In a way, differentiation produces functions whose properties are a subset of the original function, but integration produces functions whose properties expand upon the original function.
It's something pretty close to this. Almost all mathematical expressions can't be integrated (up to the reader to formalise the definitions here). That is not quite the same claim as saying that the best method for solving an arbitrary integral is trial and error by systematically differentiating other expressions until you find the antiderivative.
Both are symbols. The difference is kind of subjective, and has to do with how you treat the symbol. Do you just carry it through your derivation, or are you interested in what happens when you feed it specific values?
But I believe your objection is valid. And to be honest I got all the way through a college math major by just treating everything as symbol manipulation.
Caring about numerical values was for my other major, physics.
> Today’s article is going to discuss an obscure but powerful integration technique most commonly known as differentiation under the integral sign, but occasionally referred to as “Feynman’s technique” due to his popularization of this technique in his book, and properly known as the Leibniz Integral Rule.
It is that right integral that is equal to -pi (1+a^2)/(1-a^2).
So when you add you get:
pi/a - 1/a (1-a^2)/(1+a^2) * (-pi (1+a^2)/(1-a^2)) = pi /a - (-pi/a) = 2 pi/a
Differentiation Under Integral Sign (2015) [pdf] - https://news.ycombinator.com/item?id=26123750 - Feb 2021 (59 comments)
Feynman's Integral Trick - https://news.ycombinator.com/item?id=26040353 - Feb 2021 (6 comments)
Richard Feynman's Integral Trick - https://news.ycombinator.com/item?id=21055728 - Sept 2019 (8 comments)
Richard Feynman's Integral Trick - https://news.ycombinator.com/item?id=17558752 - July 2018 (35 comments)
It's great that we have the tricks we have, but at the same time most nontrivial integrals are just impenetrable regardless. Any demonstration of integration techniques you find will be on an integrand that is amenable to these techniques, and will only show the straight path to the solution, not the process of finding that path. I hate integrals!
$\alpha = e^{\pm ix}$
$log((\alpha - e^{ix})(\alpha - e^{-ix}))$
$2log(\alpha) + log(1 - \frac{e^{ix}}{\alpha}) + log(1 - \frac{e^{-ix}}{\alpha})$
$log(1 - x) = -\sum_{n=1}^{\infty} \frac{x^n}{n}$
$2log(\alpha) - \sum\frac{e^{inx}}{n\alpha^{n}}-\sum\frac{e^{-inx}}{n\alpha^{n}}$
$2log(\alpha) -2\sum \frac{cos(nx)}{n\alpha^n}$
Leaving just the integral of $2log(\alpha)$ which is just $2\pi log(\alpha)$
The reason, of course, is that this integral is not randomly chosen -- it represents the two-dimensional coulomb potential (log(r)) of the sphere (circle) of radius 1 at distance alpha from the center. By when point alpha is inside the circle , the potential is constant (or zero -- no force) . When alpha is outside, the potential is log(r) as if all the mass of a circle is at its center. The expression under the log in the integral is just (square of ) the distance between the point alpha and point on a unit circle.
beyond tricks -- the physical reason for the singular behavior of this integral is gauss theorem for coulomb potential . so no magic.