Mathematical integration without calculus
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- เผยแพร่เมื่อ 19 ต.ค. 2024
- I show how to convert an area measurement problem into a mass measurement problem that is easier to solve. In general, this idea of converting measurement problems into different spaces is very powerful, and may prove useful in the future.
Good examples. In about 1981 in my geochemistry class the Professor gave use a problem to work-out the partial pressure of oxygen (i.e. oxygen fugacity) of some high temp/press mineral reaction. The result was an insoluble polynomial (at least by me) that graphed as a decaying curve asymptotically approaching y=0. Anyway, most folks plotted the function on 8.5x11 notebook paper and weighed the area as you showed. Well for me, propagated by my ADHD, 8.5x11 just would not do. So, I bought a 100 foot long piece of butcher paper and using calipers and tape measure made my 100 foot area under curve. As I recall, my result was not the most accurate, but needless to say it was the most impressive. Cheers, Mark
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My goodness. Well, yes my comment was, “me me me me me.” Ben’s video reminded ‘ME’ about an experience ‘I’ had 30+ years ago and so ‘I’ relayed it as what 'I' thought was a funny/fun story relates to Ben’s demonstration. My intent was not to be bombastic and/or braggadocious, although I admit it could sound a bit like that. Cheers, Mark
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Thanks for that. Cheers, Mark
Mark Beeunas ***** I'm really glad that my viewers are able to have civil conversations. The comments section on my videos tend to be far above the average internet discourse, and I appreciate everyone who makes an effort to be reasonable. +1
Gotta ask my professor if i can bring a balance to the next exam.
Perhaps the most important application of methods like this is in education. Kids' brains are different and a certain portion of the population will comprehend concepts more readily using physical representations than by using equations only. Presenting material in a variety of ways including physical examples helps to keep more students interested in math and show people who otherwise might quit that there is a science of applied math called engineering...
Very clever. Although I suspect you increased your error in your M/A constant by writing pencil on the sheets of paper. Since there is roughly the same amount of graphite on all sheets of paper, this will increase the M/A ratio for smaller sheets - which is exactly what you notice for the 5x5 sheet. Still very clever. I enjoyed it :)
That was one of my first thoughts when I saw the writing on the papers. If you're weighing to get small values, you want as little "contaminant" on the paper as possible, and graphite would certainly be a contaminant.
You sir are a true nerd, in the best possible way.
We used this method back in the 70s on result plots from our Gas Chromatographs. Perkin Elmer had just released the first electronic integrator but at over $10,000 per ( we had about a dozend GCs in the lab) this was out of the budget. So either manual triangulation or Scissors and a sensitive lab scale where the tools of the trade :-) Tks for waking up some good memories!
I think you made the best case for it at the end. The reason that calculus is often useless to us in real life is because we have only collected real datum points, which can't possibly reveal to us their overarching pattern (function). Two points define a line, and three a plane, but even 250 samples that look like they sit on a certain curve don't prove with certainty that we have the function we're looking for. Rare is the case where we already have the actual function.
I spend my time arguing for the usefulness of mathematics, but only to a certain point; although I find it wondrous and beautiful beyond a certain level of abstraction, to me, that place fits more in the realm of philosophy than it does as a useful tool. Calculus sits on this threshold.
Mathematics goes to places that we should use caution while treading on. We have no reason to suppose that our system, while so perfect in its description of the universe that we see, will work in every possible extension. We assume multiple dimensions as we proceed, and we invent wild physical theory- and such is fascinating- but we must remember that it is all built on the assumption that our mathematics is a valid tool for every approach. The part that scares me is that it can produce apparent contradictions using only algebra, so count me as a skeptic when I hear those wild speculations about string theory on television. Of course, all of this is probably just my own ignorance making this rant, but I still like to see a problem solved with geometry- it just feels so real;)
Which contradiction can you produce with only algebra?
Math is not only about functions
This seems to be written by someone who has not studied much "higher maths". If that is what you think mathematics are all about, you have another thing coming.
@@camrouxbg Are you sure you're replying to the right comment? That makes little sense. At least give me a reason for such a quick and total dismissal. It's fair to say that I've sacrificed more than my share to the study; if the result is that I now sound general on its reflection, I take it as a compliment. Concept is the hardest skill of all to develop in mathematics.
And here is an example of (apparent) algebraic contradiction:
√-i=√-i
√(i/-i)=√(-i/i)
(√i)/(√-i)=(√-i)/(√i)
(√i)(√i)=(√-i)(√-i)
i=-i
Search out Martin Gardner for a discussion on this topic.
@@pocket83 I think your contradiction comes the fact that the root function is derived from the complex logarithm which is multivalued so you cant just chuck numbers around like they are real. The case you make that calculus is useless is pretty ridiculous, its used all the time, if not directly it is essential to make models and tools for all kinds of stuff. @Brad Camroux is definitely correct in making the assumption that you know no math since even a year in university would show you the real world usefulness of way more abstract mathematics than basic calculus.
The best example of this type that comes to mind is volume calculation by submerging complex shaped objects in water.
Thats what I was thinking of.
And what do you do if you have to measure volume of an object that absorbs water or other liquids? Use fine grained sand instead! :) I've seen that in use in food research institute, for measuring volume of e.g. bread.
Eureka!
+Kristoff B use mercury
H Mercury has quite high density and many objects would flow on top of it. Also, it's toxic.
How about a completely mechanical automatic integrator - roll of paper, cutting blade (or laser), balance....
Give this a look!
/watch?v=s1i-dnAH9Y4
geonerd Dang, that was a really good explanation.
geonerd Thanks for the share!
There are such things which were used by surveyors. They are called "planimeter"
Nice!
This reminds me of the video here on TH-cam about the old mechanical gunnery-control systems on battleships that solved gunnery target problems using all kinds of cams, disks and gizmos. Very elegant and cool.
Excellent video on solving problems by converting them into a different space. Or, as I like to call it, thinking outside the box.
This was the method of choice for finding the area under peaks in chromatography, in the time before mechanical (disc) integrators (which was before it all went electronic).
Using this very method, I was able to demonstrate that a chromatograph of TCPP isomers produced by ICI in Belgium (who were claiming that their chromatogram showed that their product had a superior isomer ratio) was produced by overloading the amplifier which gave the wrong results - they had assumed that they could get the main peaks off scale and it would all still be counted, ie, not clipped by the amplifier.
I photocopied their chromatogram onto some uniform paper, cut it out and weighed the peaks (on a four figure balance (down to 0.1mg)) and came to the same result as they did, demonstrating that their results had the main peak clipped and therefore their real isomer ratio was not as good as they claimed. (Later on, a sample of their product put through our chromatograph showed that it was worse than ours - based on their definition of what was better or what was worse)
I loved every second of reading that.
Another nice video. brings back old memories. we used the cut an weigh technique in the lab with gas and liquid chromatographs and chart recorders. the paper was fairly expensive since it needed to be very uniform for weighing. you could get very precise replicates being careful with your cutting. Not a very enjoyable task when you had dozens of samples to run with several peaks to cut out per run.
You could find an approximation of pi by weighing a circle of paper and working backwards from the measured area, dividing out the radius ^2
"Integration is just fancy addition"... just read that in a comment below... love it :-) If integration was explained in more tangible ways such as this video I bet more people would understand calculus.
When I was a kid in Blacksburg VA, my best friend's father, a VT biologist, used an analytical balance to integrate graphed data because it was quicker and simpler than the available electronic means of doing so.
Finally, a video from Applied Science that I can understand!
Years ago when designing paper gliders I weighed many sheets of paper to allow calculating a reasonable value for weight per square unit. When calculating wing loading, I would often approximate area of a given wing in calculations too... but here I see you could use a delicate scale to calculate area quickly without all the numerical integration.
As a programer, I occasionally find if a concept is difficult to express in one language, sometimes it makes more sense to invent a language where that concept is very natural to express. Then all I need to do is implement that new language. So I've converted the problem from one of expressibility into interpretation.
I am glad you build on history. This reminds me of the numberphile videos about Archimedes and the voulme of a sphere and a parabola.
You didn't account for the weight of the graphite in his calibration measurements. That's why mass over area increased for the smaller papers, because they had about the same amount of graphite on each piece regardless of its size.
Try re-doing your calibration without labeling the paper, OR re-doing your math for a linear equation with non-zero Y intercept.
Perhaps, but the paper is not uniform either and any scale used to weigh them already has a precision limit and error associated with it and will never result in a 100% accurate measurement.
He said that he assumes the paper is uniform (which it is not) and that the scale has a precision of 2mg. That in itself would already easily account for the 2% variation given the measured weights of the papers we see.
Or maybe precision of the cut.
2:42
My heart sank a little because that was the "OK" you say at the end of your videos.
In my head I instantly was like, WAIT! ITS OVER!? WHAT!?!?
But you kept going and I checked the remaining time.
All good! All good!
Moving problems into a domain you are expert in really simplifies things when it is possible. I liked the usage for the micro scale. I suspect thinner paper would improve accuracy even higher.
We used to do this sort of thing all the time when I was in undergrad (late 90's). It's handy for analog gas chromatographs, for instance, where the area under a peak is proportional to its percent abundance in the mixture that is being analyzed.
Very neat, especially the measurement of functions that don't follow any particular expression.
Love it Ben - simple is often so elegant. Just wish I could remember any of my calculus from engineering days! This is easier of course. :)
This is how my father described doing integration in chemistry class when he was in high school.
I wish my mathclasses would have been explained with examples such as this, i would have paid more attention
Cool! Makes perfect sense. What's the history on this method? If you could use very precise objects (area and weight) I wonder if the error would be closer to 0%...
Great channel. I subscribed just in hopes it will be educational by having new ways to view the sciences and how different approaches are interrelated with classic approaches. Science is like a puzzle to be solved which has enthralled me from when I was taking my toys apart at age four. And no, I did not put them together in working order until much later in life. Empirical science is always exciting too.
when I first saw the title I thought you were going into 3D but this is much more sensible
yeah, maybe use clay?..or have it upside down and hold water
i was wondering, on that small of a scale wouldn't the graphite from the pencil in which you used to write the dimensions actually throw off the measurements?
It certainly would, but the awfully simple solution to that is to not mark it with anything.
Beautiful... Just beautiful :')
Bet this could be applied in various fields beyond integration and mathematics.
I'm envious. In my life, I'm lucky to have enough energy left at the end of the day to make it home (if you call it that). And then the next day, I must devote effort to simply staying free and living with minimal dignity. It would be fun if I had time and stuff to make vids of math/engineering cleverness.
Did you try this with pieces of paper that are in equilibrium with different relative humidities?
You get surprisingly good accuracy, but maybe the mass changes on a day to day basis, depending on the weather.
Although I guess it's easy enough to make a sanity check by measuring the calibration pieces once every time before you use their data.
Cool idea. For calculating the graphs crossing the x-line you could place all negative areas on the balance, tare it, remove them and then place the positive areas. even fancier than subtracting the values manually.
At 4:21 you're wrong. Pythagoras (or archimedes? Maybe archimedes) proved with the exaustion method that the area is 1/3 of the 16x16 square.
Isn't paper susceptible to moist? The moist in the air can make the paper heavier right?
People did this in the old days? When? Because you need a high precision scale to do this. In order to deal with the precision issue, you would need heavier paper or something heavier. I think 300 years ago, they would do the Riemann (spelling?) method.
It also reminds me of what people did to find the area of a circle. Making small triangles to fit inside it. It took a while to figure out the value of tau (2*Pi) and how to use it.
My professor said this is how people computed integrals before computers got powerful enough to do numerical analysis.
Great approach and educational methodology
Interesting video, but I didn't see any error analysis to see if any of your measurements were consistent with the standard calculus method of computing area.
um i'm just concerned because the "calibrator" pieces of paper had ink on them and might have affected the mass measurements?
If you're drawing straight lines between them anyway, isn't it basically a Riemann sum?
Have you tried to measure mass of same sheet of paper in diffrent air humidity conditions, like in sunny day vs rainy day?
To substract the negative part, you don't need to actually substract.
Just use the TAR function of the scale to reset to zero!
First you place all the negative pieces onthe scale, press TAR then measure all the positive pieces :)
When our chemistry teacher told us about this, we thought it was hilarious, turns out it's very useful though!
How much error is introduced by the mass of the graphite on the cut-out paper?
Aluminum foil would probably work pretty well for this. Much more even characteristics. Tracing and cutting the shapes might be a trick though.
Many years ago I had to measure less than half a gram of powdered pigment and the only scale I had just wasn't up to it. So I "diluted" it with an inert powder to, if I remember correctly, something like 10% concentration.
In your last example, wouldn't you use a Taylor polynomial?
This is so cool. Do you think that any markings on the paper i.e. pen or pencil marks would be the reason for the margin of error?
The mass of the smallest piece of paper was 30 grams. The error on the scale was 2 grams. That's a 6,66% error right there... he got lucky.
MultiGoban
I think you misunderstood what he was doing. He had a range of masses to calibrate his measurements. So when he got his M/A conversion for paper, it was not off of the 30mg test alone but the range of samples he made. Also, the integrations he did were using pieces of paper that were over 100mg making the error in the scale less than the error he got (most likely from not cutting off the exact line). Try it out for yourself, it's fun :)
I have my integration test tomorrow, haha :D. This is a very interesting method!
Very nice. I'm surprised how consistent the paper mass is across not only one sheet, but multiple sheets.
Mechanical computation is great - check out planimeters, Ben - the math behind them is cool !
in the second case you can do (x*y)/3
Interesting. I recently read that old synthetic aperture radars used optical processing to data.
How does your brain work?! This is awesome!
what does the constant 1.47 represent?
This is how we used to do numerical integration, without computers.
No, it isn't. No one ever did integration in such an asinine way that relies on the accuracy of a hand drawn graph and scale capable of measuring micrograms.
A 2% error in mathematics is utterly unacceptable.
war1980 When we didn't have computers to do numerical integration this was the only way. I am afraid you are incorrect.
Jarod Benowitz
Because this is how Leibniz and Newton did it? Because this is how Reimann Sums work? Because Gauss' weighted function was derived by literally weighing objects?
This process is mathematically inaccurate and unacceptable. There are far more accurate ways of measuring the area under a curve than geusstimating with a hand drawn graph, and they've been used for over 300 years. For the last 2000 years, finding the area under a curve has always been done with mathematical calculations using algorithms, not the mass of whatever object you draw the graph on.
war1980 What I think they meant to say is that this is the way integration was performed without rules of calculus not even being formulated yet.
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Except it really wasn't, as far back as Archimedes, mathematicians used a form of "integration" that was based upon the same idea as a Riemann Sum, specifically shoving as many easily calculable polygons into the area and adding them up. The idea of weighing the graphed area is pretty asinine from a mathematical perspective.
Simply brilliant!
Many props to you good sir.
Rewatching this after several years.
Why not just use numerical integration methods like the trapezoid rule or the triangle rule? They seem like less of a waste of paper and can be more accurate due to less human error
Very interesting. Thank you
may be good for teaching integration too
I failed my AP Calc test. And they didn't let me keep the cut-outs.
Ha! I love it, nice and simple! Why did I never think of this!
Because doing it in the standard way is way quicker ;-) and if the integrals are too difficult, there is software that can do it numerically, instead than going back to kindergarten with paper and scissors... it is a fun to see though!
Mattia Colombo I have not taken calculus yet, but with electronics design, and learning about PID control, I have found an interest in figuring out how to calculate integrals and derivatives easily. Especially with random curves and shapes that are not expressed mathematically.
Mattia Colombo Maybe his video was a tutorial about "out of the box thinking". Don't knock someone for explaining more then one way to do something. As far as software doing it for you he shows how it works. To many people relying on machines to do the work for them without knowing how or why it works. Great job Ben love the videos keep em coming.
metrologic5000
I was actually not trying to knock anyone for anything and besides, I loved the video too =) I was just arguing that it is not exactly practical to use as a method, in particular if you want a precise results... that been said, it is really a fun way to do things, and it has really a good deal of didactic sides I think!
Yeah, I understand! I also thought the same. 5% accuracy is not the best, but for someone who does not know the hard core math, it can work in a Pinch! As well as clear up the definition of integration!
It seems like you made an error in you m/a calculation for the 5x5 piece of paper.
Great work ..,. " thank you "
There is something wrong with your figures, you give the mass of the 5x5 (area 25) as 30 mg but the ratio m/A as 1.50.
The mass should be around 37 to give that ratio.
Massaging sample data ? Lol
It should be 1.2. :) But this is not really important, as this is fun video with no practical applications.
Павел Казаков yes the ratio would be 1.2 if you had 30/25. But it is obvious that the recorded mass is wrong, since it is not reasonable to expect that the 'density' of the paper suddenly changed for that sample compared to the calculated 'density' of the other samples.
Wow this is amazing! !
What is your degree in and what college did you go to. That Is the real question.
Is your comment meant as a grunt of derision or your declaration of ignorance. Please expound, we are all dying to hear more. Cheers, Mark
No, it's not.
WHAT!!! how does one infer ignorance or derision from this comment. I was just curious
You just jumped all of what the video was about straight to asking his credentials. And not like "YOU'RE AWESOME MAN! WHERE DID YOU STUDY!" but in a dry and unengaging way. Pardon my mistrust, but that can suggest condescension or at least critic any way you put it (specially with the unengagement present), and internet is no lacking of such.
But hey, you have the right to ask, don't you?
Perhaps I am wrong, but I learned through studying Communications that each one is responsible for making him/herself being understood. No harm intended.
You seem like an ass. I hope you understand that this is meant as a "grunt of derision" and not a "declaration of ignorance."
Through studying philosophy I learned to regard people's questions with charity. Maybe your studying communications have taught the opposite.
You don't need the integration constant when calculating a definite integral.
That's great! Thanks.
Why didn't my teachers explain it this way? Would have made my EE degree a bit clearer.
I wished you'd measured π as an example.
Thank you so much!.
Very cool!
I think ***** would approve of your methods.
Meatspace integration, I like it!
Wow, that's cool!
Genius. Simply...Fucking...Genius!!
I have always wondered why engineers want to change data you can see on a chart into a computation. But then I trained in art not math.
I feel stupid for not realizing how cool math is. Teachers in school FAILED soo badly.
did this in college with some gas chromatography graphs.
last measurement show 126. so 126/1.47≈85.71
even more close to correct answer. mind blow
That's really cool
oh man if my teacher showed me this, I wouldn't have cursed him.
so genius.
The xacto is mightier than the pen.
Kevin Spacey Sound-a-Like makes applied science videos!? Subscribed!
If you were an engineer, you might alternatively curve fit the data, and then integrate...or more simply carry out numerical integration directly!
It did inspire me. *subscribes*
Next calc exam that asks to "Show your work", is getting turned in in pieces :D
Montecarlo techniques. I think the guys who made the A-bomb came up with them.
I'm baffled by my ignorance
30/25=1.2, not 1.5
great
Micrograms, blotter paper, shapes... you grew up in the seventies didn't you?
Neat cheers