I’m taking AP Calculus in high school right now, and a thought occured to me. You can tell a lot of things about a function- the area it creates when bounded by an axis, it’s instantaneous rate of change- but can you find it’s length on an interval? You could for a line, and maybe a circle. But how about any arbitrary function? Is there a field of math that studies this?
I can think of a few applications this would be useful for, but I’ve never seen it practiced.
There is such a field of mathematics. It’s called calculus. If you don’t get to it this year you’ll see it soon enough. Enjoy.
I took calculus in high school and loved it. I think they should teach it right after algebra. Knowing calculus makes everything else easier to learn.
If you have a curve parameterized by differentiable functions (x(t), y(t)), then the length along the curve from t=a to t=b is the integral from a to b of sqrt[(x’(t))[sup]2[/sup] + (y’(t))[sup]2[/sup]]dt.
Yeah, you’ll get to it later on in your class. IIRC you don’t even have to seperate it into a parametric function.
I am 1000% with you. Calculus was, Eris help me, exciting. I’ve never looked forward to a math class like I did to calc. I guess it was the thought that you could finally do something with math.
For single variable y = f(x) type stuff, the length of the arc is the definite integral from a to b of
sqrt( 1 + [f’(x)][sup]2[/sup] )dx
So long as f(x) and its derivative are continuous.
Neat! I’m actually taking Calculus BC next year and I plan to pursue a math-related field in the future, so I’m sure I’ll come across it eventually. It seems to me like a very calculus-like idea, so I’m not suprised Calc is used.
Once you get into more than two dimensions, you need a parametric specification for curves, so it’s all like Cabbage said, except you’ll take the integral of (x’[sub]1[/sub][sup]2[/sup] + … + x’[sub]n[/sub][sup]2[/sup])[sup]1/2[/sup].
BTW I agree with enjoying Calculus. I was always a good math student leading up to Calc, but I really have excelled this year. I guess I can at least partially contribute it to the fact that I enjoy it! Actually, this year I’ve decided to become an Actuary in the future and I want to get a PhD in math. Of course this is probably speaking too soon but I think I’ve found my calling.
Funny everyone is so excited about math, I was just thinking the same thing a couple days ago. I have a math degree, but wa never very satisfied with my education (crappy school, I wouldn’t suggest anyone pursue a science related field there, don’t know about the communications side though). Anyways I always liked math, and I’ve only been out of school about two years, but my math is slipping away very fast, I was just thinking about going back to a tech school or maybe even UWMilwaukee to take a couple Advanced Calc classes, or maybe some of the math classes I never had time to take, just for the heck of it.
I was lucky enough to learn Calculus from the best math teacher I ever had. I remember the day I learned how to derive the formula for the volume of a cone. It had always seemed strange to me, ever since I was first exposed to it; I mean, where does that pesky 1/3 come from anyway? The day I found out was an epiphany. Every once in awhile I take the time to rederive it, because I still find it fascinating.
Calculus is beautiful - enjoy learning it!
Wait, what? You’re taking AP Calculus now, it’s March, you know you can calculate the area bounded by function(s), but you didn’t cover arc length?
Either we’re being whooshed, or your calc curriculum is a little spotty.
Another way to find the length of a curve is to count the amount of plank-length sized superstrings that fit along it. Since distance is meaningless at scales smaller than that, you wind up with the absolute best answer. I’ll leave it up to you all here to figure out how to do that. 
Another math geek checking in. I took all the math they offered at my high school, then in my senior year I took Calc at the local college. Then when I went to college, I took nearly every math class they offered to undergrads. I think I better take a math class next semester at the local college, I might have missed a couple subjects.
Isn’t it funny how Calc textbooks are as thick as a dictionary, but once you get beyond Calc, the books start getting thinner and thinner. Perhaps that’s because the authors of the more advanced text leave the proofs of their assertions to the reader, much to the delight of math grad students. 
What most people have been posting about the arc-length formula is generally correct. Most college courses leave the general formula to the third semester, but you find a parametrization of the curve (a formula sending some region of the real number line to the curve) and integrate the square root of the sum of the squares of the derivatives of the components over that region of the real numbers. That this is independant of the parametrization you pick is relatively deep (though easy to prove).
The problem that nobody mentions is that this assumes the curve is “nice”. The method does not work for polygons, since the derivatives aren’t defined at corners. In this case, you have to break the curve into pieces at the corners and add up the lengths.
What happens, though, if the kinks in the curve get really close together? How would you measure the length of the coastline of an island? You might try putting measuring sticks along the coast and counting the number it takes to go around, but if you used a shorter stick you might end up measuring details you missed with the longer stick. Really getting a good understanding of this notion of the length of a general curve is the province of fractal geometry.
Well this is the High School curriculum. We’re only doing right now the stuff on the AP test, and that ends with solids of revolution. Looking in my Calc book, the chapter after Volume is Arc Length. Thus, I am sure we’ll go over it once the AP test is over in the beginning of May. If not then it will definitely be one of the first things we learn next year in Calc 3.
We’ve skipped some other very common things in Calculus, like integration by parts (I taught myself this just to make things easier). We’ll learn it, but our teacher would rather get the AP stuff out of the way.
I am a High School Junior and I’ve recently become very interested in becoming an actuary. In ninth grade, my Geometry teacher (Who also has a PhD in Statistics; I’m taking AP Statistics with him next year) would often mention math-related fields, and he’d always bring up the actuarial field. He’d talk about how actuaries are some of the most satisfied, well-paid, comfortable, intelligent and needed people of any job. The only thing is, he’d never really mention what they actually did!
Anyway, I was drawn back to the field this year (I don’t remember why) and I am now very interested. I know that the majority work with insurance companies, and they set and project insurance rates.
I would say I’m a very good math student- I get A’s in AP Calculus; but most importantly I enjoy math and I want to pursue a career in it. Next year I’m taking AP Calculus 3 (BC), AP Statistics, AP Economics (also AP English and Environmental Science, but those are less to do with acturaries). I also believe I’m taking one or two math courses at a local (4-year) college this summer, and possibly some more next summer. (I really want to take a Discrete Math course, but that has less to do with a career and more that it seems interesting). In other words, I will have a LOT of math credits going into college. TONS. I’m also in Future Business Leaders of America (among other things).
My mom has a friend who works at Blue Cross (an insurance agency), and she asked a actuary coworker if I could come talk to him; not only did he say yes, but he suggested that I could talk about getting an internship over the summer with them! I am definitely pursuing that to the fullest extent.
So anyway; actuaries, tell me all about your job. I hear it is one of the most enjoyable jobs, but be specific! Tell me about the exams; when did you take them, how hard are they, what subject matter do they cover? I looked at some sample (possibly earlier) tests, and they did not seem hard at all. I could even answer a good deal right now! If I studied, do you think I could pass the first before college?
About colleges: What did you major in? Few schools have majors or extensive actuary programs (Univ. of Pennsylvania has one I know; it’s been my reach school since before I was interested in the field and now I want to go even more). Statistics seems like a logical step. How about pure Math? Business? I will have so many math credits going in to college that I could easily double major.
As for Statisticians, tell me about your jobs also. I know the term “Statistics” is SO broad, and there are millions of different types. I do know that it is generally an enjoyable job and it deals with math. Are there exams like for actuaries?
Ignore/delete the thing above! It belongs in its own thread.
Actuaries are the people who decide how much a given person’s risk of death is and, thus, how much they should be charged for insurance. Well-paid and needed: yes. Intelligent: in the same sense that everyone says “oh, you must be so smart” when they learn I majored in mathematics. Satisfied: well, that rather depends. I know that there are some actuaries floating around here who even have Ph.D.s in mathematics, but as far as I’m concerned, it’s got to be some of the most boring drudgery in any “math-related” field. Yes, it takes much more math than a CPA, but it’s still an awful lot of number-crunching.
As for the rest, I’ll let the aforementioned actual actuaries field those topics and prepare to be yelled at for my hatred of numbers.
Every once in a while I differentiate an equation from first principles.
I haven’t taken calculus, so I don’t know how to derive it, but the length of a circular arc is 2[pi]rx, where r is the radius of the circle and x is the fraction of a circle it is (e.g. 0.5 for half a circle). Even if this is technically a calculus formula, I learned it in first-year geometry in 9th grade, and I think I’d even heard it before that. Beyond circles and lines though, I can’t help you.