## 2117: "Differentiation and Integration"

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Ken_g6
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### 2117: "Differentiation and Integration"

Title text: '"Symbolic integration" is when you theatrically go through the motions of finding integrals, but the actual result you get doesn't matter because it's purely symbolic.'

I went looking for a real integration flowchart. The one I found is surprisingly similar. On the plus side it doesn't have any "???"'s. On the other hand, some of those "???"'s got replaced by swear words.

pogrmman
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### Re: 2117: "Differentiation and Integration"

As my calculus teacher in high school told us, multiple times: “you can teach a monkey to differentiate, but integration is a skill”.

I’ve always been kinda interested in how asymmetrical differentiation/antidifferentiation seems — it’s really neat. I mean, even for very, very complex functions like neural nets, it’s possible to calculate the partial derivatives without necessarily doing too much (the issue is more the fact that it’s a BIG problem than that it’s a hard one).

rmsgrey
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### Re: 2117: "Differentiation and Integration"

This comic strikes me as a little anti-anti-derivative...

More seriously though, it makes a depressingly valid point about the difficulty of integration - specifically, coming up with a suitable strategy for a given integral.

Soupspoon
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### Re: 2117: "Differentiation and Integration"

The trouble with integration is that all the different sorts tend to move into their own neighbourhoods, lower your property prices when they move in next door, never mix with the natives, start to outnumber us, can't learn the local language, switch to their own tongue to make me feel uncomfortable, take all our jobs, laze away while eating up welfare…

cellocgw
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### Re: 2117: "Differentiation and Integration"

Integration is easy: you just need a small series resistor and a big cap from signal to ground.
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Eebster the Great
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### Re: 2117: "Differentiation and Integration"

pogrmman wrote:As my calculus teacher in high school told us, multiple times: “you can teach a monkey to differentiate, but integration is a skill”.

I’ve always been kinda interested in how asymmetrical differentiation/antidifferentiation seems — it’s really neat. I mean, even for very, very complex functions like neural nets, it’s possible to calculate the partial derivatives without necessarily doing too much (the issue is more the fact that it’s a BIG problem than that it’s a hard one).

It's similar to the difference between multiplying and factoring. Multiplying is very easy because there's only one thing to try. Factoring is very difficult because there are lots of things to try. It's especially relevant for multiplying and factoring polynomials; there is no quick way you could go about trying to factor an 8th order polynomial, but if I gave you 8 factors and asked you to multiply them together, you could do it mechanically in a few minutes.

sonar1313
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### Re: 2117: "Differentiation and Integration"

Alternate title for this comic: Why It Took Me Three Years To Get an Even Remotely Decent Grade in Calculus.

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### Re: 2117: "Differentiation and Integration"

Soupspoon wrote:The trouble with integration is that all the different sorts tend to move into their own neighbourhoods, lower your property prices when they move in next door, never mix with the natives, start to outnumber us, can't learn the local language, switch to their own tongue to make me feel uncomfortable, take all our jobs, laze away while eating up welfare…

Before clicking the comic, just seeing the title, I expected the comic to be about a pun like that, and wondered what exactly "differentiation" as the inverse of sociological integration would be. Segregation? Is desegregation the same exact thing as integration?
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captHij
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### Re: 2117: "Differentiation and Integration"

The comic needs an option for "clever choice of a path in the complex plane"

RyanW707
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### Re: 2117: "Differentiation and Integration"

I recall my calculus professor explaining that Calc II was Calc I backwards, and Calc III was Calc I in two directions at once. Turned out to be pretty accurate.

Archgeek
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### Re: 2117: "Differentiation and Integration"

HeHEhEehe, you think integration is bad? Wait until you encounter differential equations, A.K.A. forensic integration. You start with a horrible crime scene of related rates, and try an absurd cookbook of techniques including polynomial roots and too much application of Euler's theorem and leaps of logic no sane mind would contemplate to figure out what family of functions could've been differentiated to cause the mess.

I like to think that with the correct initial conditions, the function responsible can ultimately be brought to justice.
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Jorpho
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### Re: 2117: "Differentiation and Integration"

I was lucky to have a really good teacher for Cal II, I think. He made it feel like a beautiful, satisfying culmination of everything that had come before.

Now, Linear Algebra – that was the one that seemed to trip everyone up. Just a big ol' mess.

And I still feel a little guilty that I never got around to figuring out complex analysis. I recall in "Surely You're Joking, Mr. Feynman", he suggests doing away with it and using differentiation under the integral sign instead, which I never figured out either. Oh well.

Mikeski
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### Re: 2117: "Differentiation and Integration"

Pfhorrest wrote:
Soupspoon wrote:The trouble with integration is that all the different sorts tend to move into their own neighbourhoods, lower your property prices when they move in next door, never mix with the natives, start to outnumber us, can't learn the local language, switch to their own tongue to make me feel uncomfortable, take all our jobs, laze away while eating up welfare…

Before clicking the comic, just seeing the title, I expected the comic to be about a pun like that, and wondered what exactly "differentiation" as the inverse of sociological integration would be. Segregation? Is desegregation the same exact thing as integration?

Integration = "melting pot". Cultural appropriation is a force for good.

Differentiation = balkanization. Cultural appropriation is a force for evil.

Qaanol
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### Re: 2117: "Differentiation and Integration"

Jorpho wrote:And I still feel a little guilty that I never got around to figuring out complex analysis. I recall in "Surely You're Joking, Mr. Feynman", he suggests doing away with it and using differentiation under the integral sign instead, which I never figured out either. Oh well.

The nitty-gritty of complex analysis can get really tedious and involved.

However, the high-level results are simply astounding:

…being differentiable *once* automatically implies being infinitely differentiable?

…a differentiable function cannot have its maximum magnitude on the interior of its domain?

…any derivative of a function at a point can be found by taking a certain integral around that point?

…closed-loop integrals can be evaluated by looking at the 1/x term of the power series at the function’s *poles*???
wee free kings

DeGuerre
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### Re: 2117: "Differentiation and Integration"

Integration: Risch algorithm. Nothing could be simpler.

djagir
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### Re: 2117: "Differentiation and Integration"

Quote from one of my engineering professors at college: "All I remember about calculus is that differentiation is easy and integration is impossible."

gcgcgcgc
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### Re: 2117: "Differentiation and Integration"

In the various forms of Path Tracing in Computer Graphics, integration isn't hard, it's just slow. Of course they cheat and do everything numerically instead, by taking random samples and turning it into a discrete sum. But it's a bit like firing a gun in random directions and hoping to hit a lightbulb (or your eyeball, if working in the opposite direction)
It would be a lot faster if true integration was easier though, and a few simplified cases are solvable and used in real-time graphics.
You can use importance sampling to speed things up, if you're careful to avoid biasing the result.

NemeSys
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### Re: 2117: "Differentiation and Integration"

If I get really stuck on an integral I just throw Gradshteyn and Ryzhik (8th edition) at it and hope for the best. That or go into a foetal position and cry myself to sleep...

The Moomin
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### Re: 2117: "Differentiation and Integration"

It was integration at university that made me realise the futility of maths.

Primary school - calculate the area of a shape by counting the number of squares and half squares it covers on the graph paper.
Middle school - learn the basic formula for calculating the area of shapes.
High school - learn integration for calculating the area beneath the curve where you know the formula.
University - calculate the area beneath a curve by modelling the area as tiny squares you can add up.

It's just unnecessary complications.
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doogly
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### Re: 2117: "Differentiation and Integration"

Once when I was teaching calc 2, the kiddos asked how to know when to use what strategy. I told them, go home and do a thousand. This was poorly received, mostly, and yet.
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Flumble
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### Re: 2117: "Differentiation and Integration"

DeGuerre wrote:Integration: Risch algorithm. Nothing could be simpler.

Lots of things are simpler, like eating a grape, but they don't get you an integral.
Thanks for the link, now I have a basic grasp of field extensions!

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### Re: 2117: "Differentiation and Integration"

This is probably my biggest gap in my calculus instruction: when to actually use these techniques. In class you lean heavily on "we were just taught this technique, so we should probably use it".

pogrmman wrote:As my calculus teacher in high school told us, multiple times: “you can teach a monkey to differentiate, but integration is a skill”.
That seems a perilous thing to say for somebody actively teaching integration.

One student failed? "You can teach a monkey to differentiate, the editorial 'you"; obviously, you (specifically) can't teach that well."
The thing about recursion problems is that they tend to contain other recursion problems.

sonar1313
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### Re: 2117: "Differentiation and Integration"

Archgeek wrote:HeHEhEehe, you think integration is bad? Wait until you encounter differential equations, A.K.A. forensic integration. You start with a horrible crime scene of related rates, and try an absurd cookbook of techniques including polynomial roots and too much application of Euler's theorem and leaps of logic no sane mind would contemplate to figure out what family of functions could've been differentiated to cause the mess.

I like to think that with the correct initial conditions, the function responsible can ultimately be brought to justice.

My computer uses precogs. Every so often it spits out a function before I even know whether or not the professor assigned any homework problems.

Eebster the Great
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### Re: 2117: "Differentiation and Integration"

DeGuerre wrote:Integration: Risch algorithm. Nothing could be simpler.

If I understand correctly, every implementation of this algorithm must fail for certain integrals because it is undecidable whether parts of the integrand ever equal zero. (This isn't specific to the Risch algorithm; it is a problem with any CAS algorithm.)

24b4Jeff
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### Re: 2117: "Differentiation and Integration"

A block is missing from the Integration half: "Consult Gradstheyn & Ryzhik". I know, I know, its a 'book' and thus totally uncool. Even though it is also available as a CD. No, it's not an audio CD, although, come to think of it, an audio CD of G&R would be excellent for evening listeing.

A brief excerpt might start with, "...and tonight we will start with section 5.22, combinations of the exponential integral function and powers. 5.2.1.1 integral from x to infinity of the exponential integral function, with argument minus beta x, times . . ." You get the idea.

Archgeek
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### Re: 2117: "Differentiation and Integration"

24b4Jeff wrote:A block is missing from the Integration half: "Consult Gradstheyn & Ryzhik". I know, I know, its a 'book' and thus totally uncool. Even though it is also available as a CD. No, it's not an audio CD, although, come to think of it, an audio CD of G&R would be excellent for evening listeing.

A brief excerpt might start with, "...and tonight we will start with section 5.22, combinations of the exponential integral function and powers. 5.2.1.1 integral from x to infinity of the exponential integral function, with argument minus beta x, times . . ." You get the idea.

Good lasers, that sounds like an unatapped ASMR genre -- dorks on youtube whispering complex mathematics texts to trigger the Auto-Somatic Mathematics Response.

edit: that blank line did not go there
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doogly
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### Re: 2117: "Differentiation and Integration"

Quizatzhaderac wrote:This is probably my biggest gap in my calculus instruction: when to actually use these techniques. In class you lean heavily on "we were just taught this technique, so we should probably use it".

pogrmman wrote:As my calculus teacher in high school told us, multiple times: “you can teach a monkey to differentiate, but integration is a skill”.
That seems a perilous thing to say for somebody actively teaching integration.

One student failed? "You can teach a monkey to differentiate, the editorial 'you"; obviously, you (specifically) can't teach that well."

That is why my calc teacher preferred the conditional, I could teach a stone to get a 5 on the ap if the stone did its homework.
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doogly
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### Re: 2117: "Differentiation and Integration"

Quizatzhaderac wrote:This is probably my biggest gap in my calculus instruction: when to actually use these techniques. In class you lean heavily on "we were just taught this technique, so we should probably use it".

pogrmman wrote:As my calculus teacher in high school told us, multiple times: “you can teach a monkey to differentiate, but integration is a skill”.
That seems a perilous thing to say for somebody actively teaching integration.

One student failed? "You can teach a monkey to differentiate, the editorial 'you"; obviously, you (specifically) can't teach that well."

That is why my calc teacher preferred the conditional, I could teach a stone to get a 5 on the ap if the stone did its homework.

Also, yes, big up to the Russian Book.
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rmsgrey
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### Re: 2117: "Differentiation and Integration"

doogly wrote:
Quizatzhaderac wrote:This is probably my biggest gap in my calculus instruction: when to actually use these techniques. In class you lean heavily on "we were just taught this technique, so we should probably use it".

pogrmman wrote:As my calculus teacher in high school told us, multiple times: “you can teach a monkey to differentiate, but integration is a skill”.
That seems a perilous thing to say for somebody actively teaching integration.

One student failed? "You can teach a monkey to differentiate, the editorial 'you"; obviously, you (specifically) can't teach that well."

That is why my calc teacher preferred the conditional, I could teach a stone to get a 5 on the ap if the stone did its homework.

Well, I could teach a stone to get a Nobel Prize if the stone did its homework.

Eebster the Great
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### Re: 2117: "Differentiation and Integration"

rmsgrey wrote:
doogly wrote:That is why my calc teacher preferred the conditional, I could teach a stone to get a 5 on the ap if the stone did its homework.

Well, I could teach a stone to get a Nobel Prize if the stone did its homework.

Silicon is kinda stone-ish, and I could program some to do homework, but I doubt you could teach it to win a Nobel. Actually, I'm pretty sure there are already silicates doing homework, so we can test this.

Pfhorrest
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### Re: 2117: "Differentiation and Integration"

Silicon dioxide has no "ish" about it, that's straight up what most of most stones are made of.
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Eebster the Great
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### Re: 2117: "Differentiation and Integration"

Pfhorrest wrote:Silicon dioxide has no "ish" about it, that's straight up what most of most stones are made of.

I am not so educated in the ways of minereroalogy as to know stone from metal or clay or whatnot. But it's a thingy and it does homework.

herdi42
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### Re: 2117: "Differentiation and Integration"

Integration is easy after you have discovered the concept of Bronstein (or Bronshtein, or whatever) integrability, as a math tutor at my university called it:

Got me through all math, physics, physical chemistry, and theoretical chemistry / quantum mechanics courses.

GlassHouses
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### Re: 2117: "Differentiation and Integration"

Pfhorrest wrote:Silicon dioxide has no "ish" about it, that's straight up what most of most stones are made of.

It's not what computer chips are made of, though. That would be highly pure silicon, and silicon is a metalloid, not a rock.

cellocgw
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### Re: 2117: "Differentiation and Integration"

Only because I can't believe we got this far without someone else posting these instructional links,

Integrate and differentiate
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sharpfang
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### Re: 2117: "Differentiation and Integration"

It's not the solution that matters, it's the activity of integration itself!

chenille
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### Re: 2117: "Differentiation and Integration"

sharpfang wrote:It's not the solution that matters, it's the activity of integration itself!

Then the real integral is the friends we made along dy?

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### Re: 2117: "Differentiation and Integration"

Let S represent "The solution"
Let I represent "Integration "
Let a(x) indicate "The activity of x"
____as such, a(I) represents "The activity of Integration
Let m(x) represent the statement "x matters"
____as such, m(S) represents "The solution matters"
____as such, m(a(I)) represents "The activity of the integration matters"

All activities have a start and end point, as such
____Let B represent the beginning point of a(I), whenever it is
____Let E represent the ending point of a(I), whenever it is
____Let U represent the starting point of the utilization of a(I)
________Note that "Utilization" includes work project, homework assignments, academic paper submission, smarmy forum posts, and many more cases.

now, let us consider the case where integration is done properly:
BEa(I)dt = S
Assuming !m(S) (as per sharpfang)
!m(∫BEa(I)dt)
!(m(∫BEa(I)dt))
!(m(B) and m(E) and m(a(I)))) by Tracy's Law of composition of meaning
!m(B) or !m(E) or !m(a(I)) by De Morgan's laws

By Kholl's law of causation, B < E < U
By the axiom of utility: m(U)
By Kholl's third law: ((E<U) and m(U)) -> m(E)
By Kholl's third law: ((B<E) and m(E)) -> m(B)

Composing demonstrated statements: m(B) and m(E) and (!m(B) or !m(E) or !m(a(I)))
simplifying: !m(a(I))
!m(S) -> !m(a(I))

If the solution doesn't matter, then the activity of integration doesn't matter.
The thing about recursion problems is that they tend to contain other recursion problems.

Pfhorrest
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### Re: 2117: "Differentiation and Integration"

GlassHouses wrote:
Pfhorrest wrote:Silicon dioxide has no "ish" about it, that's straight up what most of most stones are made of.

It's not what computer chips are made of, though. That would be highly pure silicon, and silicon is a metalloid, not a rock.

Then all of the jokes about computers being made out of sand fall apart, because sand is mostly silicon dioxide.
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Archgeek
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### Re: 2117: "Differentiation and Integration"

Pfhorrest wrote:
GlassHouses wrote:
Pfhorrest wrote:Silicon dioxide has no "ish" about it, that's straight up what most of most stones are made of.

It's not what computer chips are made of, though. That would be highly pure silicon, and silicon is a metalloid, not a rock.

Then all of the jokes about computers being made out of sand fall apart, because sand is mostly silicon dioxide.

Chips contain plenty of oxide, though. The wafer-thin slices of hilariously pure silicon crystal are very commonly doped with its oxide in order to provide the p-n and n-p junctions that make useful diodes and transistors happen. So at the very least, they certainly contain sand.
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