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Going into university, how many of you know how to integrate a function?

A photo of Phase Phase
This is mainly aimed at Ontario students due to our dumbed down curriculum, but students from other provinces are free to answer. How many of you know how to integrate a function for those of you who have finished calculus or the calculus portion during this semester? I plan on teaching myself this and practicing it over the summer. If you decide to answer, please state which province you currently reside in.
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A photo of greygoose greygoose
Coming from Alberta (though I am aware they recently dumbed down the math curriculum, though maybe they left Calculus alone), I knew how to integrate out of high school. I also took AP Calculus (AB, the only one offered), which really solidified the ability of basic integration since they focus a *lot* on doing and the larger half of the course is on integration.

Aaaanyhow, I have some nicely typeset notes from the second half of first year calculus which is an introduction to integration, assuming very little prior knowledge (though you can probably skip most of the stuff that does since it's all proofs of theorems basically). If you like I can send. They probably still have some mistakes, I was planning on vetting them this summer and typesetting my first term notes, but it hasn't been something I've got around to yet. They do kind of assume one is aware of integration, and that one has had an introduction to analysis, but you can probably wade through all that and find that the theory is interesting. I was planning on adding a few more methods of integration to the notes as well eventually.
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A photo of LRooke LRooke
Hell yeah I can integrate a function. As well as how to calculate the area bounded between curves and lines. And how to derive velocity and displacement equations. God bless the mighty GCSE system.
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A photo of chatmike chatmike
I do not since it is not part of the ontario curriculum (ridiculous), so i will be teaching myself in the summer like you.
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A photo of Yaroslav64 Yaroslav64
Someone care to explain what that even means? I'm doing the calculus part of the course and we just wrapped up optimization, I'm pretty sure we're not doing integrating functions.
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A photo of LRooke LRooke

@Yaroslav64 wrote
Someone care to explain what that even means? I'm doing the calculus part of the course and we just wrapped up optimization, I'm pretty sure we're not doing integrating functions.




Integration is the reverse of Differentiation.
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A photo of JNBirDy JNBirDy
I didn't know how to integrate out of high school... I did fine in university calculus.
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A photo of North North
I know how. But that's because I did IB. Even in standard level it covers calculus from Limits to volume of revolutions.
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A photo of Nyx Nyx
I don't either, and i'm going to do it over the summer too. Why not make first year as easy as possible?
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A photo of greygoose greygoose

@LRooke wrote
Integration is the reverse of Differentiation.



*Inverse, thank you very much. :)

I would be cautious in teaching yourself this stuff over the summer. You may misunderstand things or teach yourself wrong, and then have bad habits to break in the fall.

Math majors don't even cover integration until second term anyways (or maybe a small amount at the end of first term in the normal sections, I believe), and it's not really assumed for science majors in their first term, though I believe they will learn in then. I don't think it's assumed for engineering either, to be honest. How could it be? It's not part of the curriculum for the vast majority of high school students coming into Waterloo.
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A photo of AsianFail93 AsianFail93
Do I know how to do it properly? nope
I kinda know how to do it with very simple equations
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A photo of Phase Phase

@greygoose wrote
Coming from Alberta (though I am aware they recently dumbed down the math curriculum, though maybe they left Calculus alone), I knew how to integrate out of high school. I also took AP Calculus (AB, the only one offered), which really solidified the ability of basic integration since they focus a *lot* on doing and the larger half of the course is on integration.

Aaaanyhow, I have some nicely typeset notes from the second half of first year calculus which is an introduction to integration, assuming very little prior knowledge (though you can probably skip most of the stuff that does since it's all proofs of theorems basically). If you like I can send. They probably still have some mistakes, I was planning on vetting them this summer and typesetting my first term notes, but it hasn't been something I've got around to yet. They do kind of assume one is aware of integration, and that one has had an introduction to analysis, but you can probably wade through all that and find that the theory is interesting. I was planning on adding a few more methods of integration to the notes as well eventually.


I'd appreciate it if you can send, thanks. :cheers:

Also, what tips would you have to transition from computational-based mathematics (high-school) to proof-based mathematics?
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A photo of JNBirDy JNBirDy
If you guys want to get a head start on university calculus check this out:

http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/

It's an online course offered by MIT.
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A photo of pandasbox pandasbox
I took AP calc BC and now I can integrate a function with my eyes closed (figure of speech, I prolly can't actually integrate it if I don't see it). But what was more useful for me to learn was infinite series, because it helped me understand all the physics approximations, and they don't even teach that in calc AB. If you don't know it yet, I highly recommend previewing it from a first year calc textbook.
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A photo of greygoose greygoose

@Phase wrote
I'd appreciate it if you can send, thanks. :cheers:

Also, what tips would you have to transition from computational-based mathematics (high-school) to proof-based mathematics?



Argh, now I'm going to have to typeset this stuff. Well I mean, my second term notes are already (the ones with the integration), but they lack pretty pictures, full proofs, etc. Basically, they are not ready for the eyes of math-virgins. Was hoping to get them up to snuff this summer.

The thing is, as I said, integration is more of a side note in my notes because it was kind of assumed we knew how being the advanced section. We had a single integration question on the final.

Basically what I'm saying is, I'd be cautious in sharing these with a bunch of pre-frosh because it's *not* typical first year stuff and I don't want to scare you. It wouldn't be expected of you.

I also have no idea what program you are going in. For some reason I just assumed Waterloo. And even then, it'd depend *what* program at Waterloo.

Math majors get a lot of the "proof-based" stuff, but mostly the more abstract/theoretical mathematicians. The more applied programs (ex. many financial ones) are much more doing and much less understanding, which is unfortunate, but the emphasis on proofs there is far lower. The normal sections don't really understand the rigor of proofs until 2nd year or later, whereas the advanced sections are expected to "get it" from day 1. So you're really comparing a lot of different things.

And then engineers and science majors (perhaps barring theoretical physicists) have very applied math and need to know a lot of integration "doing" but have much less need for understanding at the undergrad level.

But if you are going into a more abstract program, really the only way to learn proofs is doing them. A lot. On assignments. Twisting the brain. Going to many many office hours. I don't think a week went by first term where I *didn't* go to office hours for calculus. Learning how to do your own proofs are hard to wrap one's head around the first time, and it took me until about 6 weeks into my first term before I really understood. By the end of the term I was able to prove some pretty heavy stuff just off the top of my head for exams (no memorization). The normal classes are not expected to do this--only proving simple things or recalling memorized proofs. So... it depends, basically. Provide me with more background and I can answer better.
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A photo of Phase Phase

@JNBirDy wrote
If you guys want to get a head start on university calculus check this out:

http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/

It's an online course offered by MIT.


Thanks so much. This is great! :cheers:

I wonder, do they offer the same materials for a head start in linear algebra and analysis?
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A photo of Zion Zion
I know how. It's part of the standard curriculum in BC for Calculus. I thought it was the hardest unit though.
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A photo of Phase Phase

@greygoose wrote

@Phase wrote
I'd appreciate it if you can send, thanks. :cheers:

Also, what tips would you have to transition from computational-based mathematics (high-school) to proof-based mathematics?



Argh, now I'm going to have to typeset this stuff. Well I mean, my second term notes are already (the ones with the integration), but they lack pretty pictures, full proofs, etc. Basically, they are not ready for the eyes of math-virgins. Was hoping to get them up to snuff this summer.

The thing is, as I said, integration is more of a side note in my notes because it was kind of assumed we knew how being the advanced section. We had a single integration question on the final.

Basically what I'm saying is, I'd be cautious in sharing these with a bunch of pre-frosh because it's *not* typical first year stuff and I don't want to scare you. It wouldn't be expected of you.

I also have no idea what program you are going in. For some reason I just assumed Waterloo. And even then, it'd depend *what* program at Waterloo.

Math majors get a lot of the "proof-based" stuff, but mostly the more abstract/theoretical mathematicians. The more applied programs (ex. many financial ones) are much more doing and much less understanding, which is unfortunate, but the emphasis on proofs there is far lower. The normal sections don't really understand the rigor of proofs until 2nd year or later, whereas the advanced sections are expected to "get it" from day 1. So you're really comparing a lot of different things.

And then engineers and science majors (perhaps barring theoretical physicists) have very applied math and need to know a lot of integration "doing" but have much less need for understanding at the undergrad level.

But if you are going into a more abstract program, really the only way to learn proofs is doing them. A lot. On assignments. Twisting the brain. Going to many many office hours. I don't think a week went by first term where I *didn't* go to office hours for calculus. Learning how to do your own proofs are hard to wrap one's head around the first time, and it took me until about 6 weeks into my first term before I really understood. By the end of the term I was able to prove some pretty heavy stuff just off the top of my head for exams (no memorization). The normal classes are not expected to do this--only proving simple things or recalling memorized proofs. So... it depends, basically. Provide me with more background and I can answer better.


I am going into mathematical physics specialist at UofT, so I am probably going to be taking real analysis, topology, number theory, etc. when I am in my third year along with other pure math majors.

Also, if you don't mind me asking, how proof based is a first year analysis course? Would you advise against a first year math students from taking it?
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A photo of greygoose greygoose

@Phase wrote
I am going into mathematical physics specialist at UofT, so I am probably going to be taking real analysis, topology, number theory, etc. when I am in my third year along with other pure math majors.

Also, if you don't mind me asking, how proof based is a first year analysis course? Would you advise against a first year math students from taking it?



A proper first year analysis course is 90% proof-based 10% "doing". By "doing" I mean taking derivatives, finding minima/maxima, etc.

I think there were like all of 5 assignment questions on 10 assignments first term that weren't proofs. Our final was like maybe 80% proofs. It was a really excellent course. Second term calc I thought had too much emphasis on doing and not enough on theory and it frustrated me a lot.

These are all rough estimates of course...

I would love to share my notes because they are from a true introduction to analysis course, but they're simply not ready yet :( They're not typeset and they lack many proofs.

What I caaaan do is start working on them this weekend and publish them bit by bit. If that interests you. A real introduction to analysis, if you will.
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A photo of Phase Phase

@greygoose wrote
A proper first year analysis course is 90% proof-based 10% "doing". By "doing" I mean taking derivatives, finding minima/maxima, etc.

I think there were like all of 5 assignment questions on 10 assignments first term that weren't proofs. Our final was like maybe 80% proofs. It was a really excellent course. Second term calc I thought had too much emphasis on doing and not enough on theory and it frustrated me a lot.

These are all rough estimates of course...

I would love to share my notes because they are from a true introduction to analysis course, but they're simply not ready yet :( They're not typeset and they lack many proofs.

What I caaaan do is start working on them this weekend and publish them bit by bit. If that interests you. A real introduction to analysis, if you will.


90% proof based? Woah. I imagine there would be a high drop-out rate for those students who decide to take it in their first year? So, what kind of content do they go over? I know real analysis is basically proof-based, rigorous version of calculus, so analysis would be the rigorous and proof-based equivalent of pre-calculus?

I totally understand where you are coming from. My calculus course was basically "plug n chug" and our teacher refused to prove any of the rules and theorems and just told us to memorize them. Since the majority of my class were planning on engineering or business, they didn't really care about proofs but only about the solution. I kind of wished there was an option from grade 9 to start doing proof-based math or computational math, since the latter would be much useful to aspiring engineers and business types while the former will be extremely helpful and useful to aspiring math and physics majors. After taking a math course that is proof-based, what do you think you could have done better to prepare for it?

I'd be grateful if you would share your notes with me. Thanks so much, you're a life-saver.
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A photo of noxx98 noxx98

@Phase wrote
I plan on teaching myself this and practicing it over the summer.


This is dumb and a waste of time. You won't do integration till the end of your first semester, and its pretty easy to get the hang of.

Go out and get drunk - much better use of your time.
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A photo of LRooke LRooke
I bet I have my original Integration notes somewhere. I'll scan and host it for y'all to download.
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A photo of merm123 merm123

@LRooke wrote
I bet I have my original Integration notes somewhere. I'll scan and host it for y'all to download.



Awesome :)

Our calculus class has yet to learn that..pretty sure our teacher said he would teach it to us though cause there's a ton of mathies/engineering people in our class.
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A photo of greygoose greygoose

@Phase wrote
90% proof based? Woah. I imagine there would be a high drop-out rate for those students who decide to take it in their first year? So, what kind of content do they go over? I know real analysis is basically proof-based, rigorous version of calculus, so analysis would be the rigorous and proof-based equivalent of pre-calculus?



I plan on having a chunk of my notes texxed out tonight, so you'll be able to see the introduction. Note that I did qualify with "a proper introductory analysis course"; I highly doubt the majority of them would have such rigor. At Waterloo, this course (Math 147) is optional. It usually starts with about 100 interested people in the first 2 weeks, drops to 70 after that, and stabilizes around 50 after the midterm. But these people aren't failing out or anything, they're just dropping to the normal stream. It's an optional thing thanks to the difficulty, but very rewarding. In the end, only the people that belong in the class are taking it, so while some people can come out with lower marks, I have never heard of a person who put in the required work and failed.


@Phase wrote
I totally understand where you are coming from. My calculus course was basically "plug n chug" and our teacher refused to prove any of the rules and theorems and just told us to memorize them. Since the majority of my class were planning on engineering or business, they didn't really care about proofs but only about the solution. I kind of wished there was an option from grade 9 to start doing proof-based math or computational math, since the latter would be much useful to aspiring engineers and business types while the former will be extremely helpful and useful to aspiring math and physics majors. After taking a math course that is proof-based, what do you think you could have done better to prepare for it?



I don't think there was anything I could have done to prepare for it. It was like being beat over the head with a 2 by 4. Seriously, it's a very different way of thinking, and takes a long time to get used to. It takes a good 6 weeks to get there.

You know, I really think proper, rigorous mathematics could be taught at the high school level, but the curriculums just don't allow for it. There's a lot of silly stuff I don't think is particularly useful, and a lot of stuff I think really should be in there (at minimum, available for an advanced curriculum). By the time you get to your twelfth year and start learning calculus, to just jump into the proper foundation for proofs maintaining a high school pace is simply not possible. And the high school level proofs they might show you for some of the theorems are based on enormous assumptions (I'm looking at you, handwavey proof of the second part of the fundamental theorem of calculus! >.>).


@Phase wrote
I'd be grateful if you would share your notes with me. Thanks so much, you're a life-saver.



It is my pleasure! I planned on starting on them now.
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A photo of ktel ktel
Sigh I miss math. Proofs made so much sense to me and I would much rather know how to derive something than have to memorize it. It's been a little while since I've done math proofs and some of the papers I have to read now make me wish it hadn't been so long!

I took IB Math (Standard Level) in high school, which was a while ago, and we just barely touched on integration. It should be easy enough to teach yourself the basics of how to do it (it really only requires a lot of practice, knowing what to recognize). I would hold off on things like integration by parts
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A photo of CanadaBoy CanadaBoy
Took 1st term University Calculus in High School, so I know how to integrate, find the area between curves, bounded by a curve, average value, work, etc.
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