WHAT IS CALCULUS?
Just the word spooks the horses and sends villagers clamoring for their pitchforks.
A system of arcane squiggles you briefly glimpsed that one time you went to a Bad Place on Wikipedia and had the sanity burned from your eye holes.
The Oz behind the curtain. The lands beyond the Wardrobe. The fierce doorman that decides who gets in Club Science to boogie with the hotties.
The monster in the closet.
Stay with us -- we promise this is headed somewhere.
What is Calculus?
Remember all that talk from your algebra teacher about how important factoring quadratics would be in everyday life? Lies. All of it.
Calculus is where the rubber really meets the road.
Sure, algebra is important in the sense that it's necessary to do calculus, just as arithmetic is important to do algebra. And English is important to read your math books. And so on. A line of argument that can be traced all the way back to finger-painting and eating paste. But LabKitty lives in the now (or the meow, as it were).
So we'd like to describe the wild beast that is calculus to the uninitiated.
The question is not why, but rather: Why not?
As a society we appreciate the beauty of a well-thrown spiral into the end zone, or the gushing allure of a good hockey fight. We hazard some of you appreciate painting without being able to paint or music without being able to toot. Why not, then, calculus? The Crown Jewel of Science, it has been called probably by someone at some point. Let's see what the hoopla is all about. Let's head on down to the dealership and kick the tires. Let's give our Harsh Mistress a poke.
No, we're not going to teach you to DO calculus. That would be a tall order. There's a reason why university calculus takes two years and why the calculus section in books stores is big enough to be seen from space. Our modest goal here is to tell you what calculus IS, not how to do it. The gist. The thrust. What the French call a certain I don't know what. Just enough so that you'll know it when you see it. Like crochet or Jai Alai. Like sighting a famous starlet buying half-n-half in the 7-11 at 3 AM. There's lots of details we'll be skipping.
So if you've ever thought: what is that calculus stuff all the cool kids seem to be going on about?, read on. Heck, perhaps once bitten you will decide to embiggen your brain with actual calculus knowledge. If so, we'll point you in some textbook directions here and there. But that's not necessary to appreciate the beauty, the pageantry, the spectacle. As they say on Vulcan, spectacle is the beginning of wisdom.
The first thing you need to know is there is a great rift in calculus. A split, a cleft, a chasm. And so our story begins there.
Calculus - Two sides of one coin
The coin of the realm
Vampires and Werewolves. Imperials and Stormcloaks. Jocks and Nerds. Aliens and Predators. Persia and Sparta. Romulans and Capulets. Christians and Hogwarts.
The great realm of calculus is also riven in two. Its halves are: differential calculus and integral calculus.
In the simplest introductory language we can conjure, one might say differential calculus is the study of rates-of-change, and integral calculus is the study of areas. Neither of those descriptions will mean much to you now, but they will soon. We mention them here only to point out these are two sides of the same coin. That may seem preposterous. For what does a rate of change have to do with finding an area? Or, put another way, what does finding an area have to do with a rate of change? That's the weird part of calculus. That's the part that's by no means obvious. And that's why the person (a couple people, really) who first figured it out is famous forever.
Having teased you with that, however, do not let it trouble your mind too much. For although connecting the two halves of calculus is a bit of a magic trick, that comes later. We dare say each half in itself is rather simple. And so let us examine the differential calculus and the integral calculus each in turn.
Begin your tour here
Integral calculus is usually taught after differential calculus in a typical calculus course, but we're going to do it first. Why? Because we're rebels (chicks dig rebels). Actually, because we're so confident that the concepts of calculus are simple -- even if the details are not -- that it doesn't matter in which order we take them on.
We're not going to pretend this is hard. Let's jump right in to the gaping maw of an example.
We have a circle with radius (r) equal to one. We want to find the area of the circle. Why is not important. But we can't remember the handy formula (area = pi-r-squared). What to do?
Well, we may not know how to compute the area of a circle, but we're not daft: we do know how to compute the area of simpler shapes. A square, for example. So here's the plan: (1) draw a square around the circle. (2) tile the square using a collection of smaller squares, like tiling a bathroom floor, (3) throw out the tiles that are outside the circle, (4) count the number of tiles left. Call this number "N." Logic tells us that the area of the circle is equal to N x A, where "A" is the area of one tile.
Now, this number is only an approximation. Why? Because square tiles don't fit perfectly inside a circle (see accompanything figure). The solution? Use more and more smaller and smaller tiles.
Here we turn to the computer for help. In the table below are actual real results of simulated tiling (this can be whipped up on a spreadsheet or some other means of gronkulation -- we used MATLAB). The first column refers to the panel in the accompanying figure (a,b,c, or d) that shows the tiling, then comes the tile size, the tile area, N for the tiling, and the area approximation obtained.
The circle that we're working on has a radius of 1, so we know the true area is about 3.14. The first two approximations are not so hot. The single tile in (a) and all four tiles in (b) overlap the circle and so were counted in the approximation. The sum total area in both cases is 4. Version (c) uses 32 squares (we throw out the tiles in the four corners — that’s why they’re crossed out) and the area approximation begins to improve. In version (d) the tiling is starting to look a little like a circle, and the approximation is better still (although still not outstanding).
Now comes the calculus part.
Area Approximation II
We ask: what would happen if we play this game forever? That is, we keep using smaller and smaller squares until they are smaller than the smallest square you can image. Smaller than dust. Smaller than specks. Smaller than atoms. Small. Each step in the process -- each time we play the game --- is nothing new. We're still just counting squares. There just lots more of them. What's new is the idea of continuing the game forever. Is there a way to tell where we will wind up, even if we don't really ever get there? Calculus says: yes.
Here, a reasonable person might say: That's redonkulus! If we keep adding up more and more smaller and smaller squares, won't we eventually be adding up infinitely many nothings? How can that result be anything other than zero? Or is it infinity? It certainly isn't a regular number like π, no? Grrr!
A fair question. We again turn to the computer -- if not for an answer then at least for some convincing. In the lower half of the accompanying figure, we went crazy berserk with tiling. The numbers on the x-axis is the size of the tiling squares (decreasing to the right) and on the y-axis is the area approximation obtained using a tiling of that size. The first four data points are four tilings we discussed above. The additional data points show the results of using smaller and smaller tilings. We note two things:
(1) As we shrink the size of the tiles, the approximations obtained neither shrink to zero nor run off to infinity. Apparently tiny divided by tiny remains an ordinary number!
(2) We get a more accurate answer if we use smaller tiles, yes, but there's less improvement with each additional refinement. That is to say there is a sort of diminishing return to the whole business. The tiniest tiling we tried (size: 0.0001 - not shown on graph) used more than 300 million squares and took 30 minutes for our little computer to crank though. Yet, the result was not spectacularly better than the last data point shown on the graph (size: 0.001), which used only about 3 million. We say the results are converging; specifically they are converging to π (about 3.14) which we know is the correct answer.
Now you may say: that's a lot of squares, Kitty, but that's still not "forever." How do you know that Something Bad doesn't happen eventually? Congratulations! You are starting to think like a calculus student. But we ask you: what goes wrong? Or, more to the point, when goes wrong? We know the approximations are well behaved in version a, version b, and so on. If something goes wrong eventually, there must be a version in which something went from good to bad. What happened between that version and the version that came before?
Admittedly, that is not an answer. But it is the beginning of one. Going further would require details, the kind we warned you that we're not going to get into. But we hope the overall idea is clear: (1) tile circle. (2) buy smaller tiles. (3) go to step #1. Repeat until the numbers converge on a result.
Meet the Integral
Of course you can't play the tile game forever, even on a computer. But in a calculus course you learn techniques for finding out what the answer WOULD be if you DID. The quantity you get if you spend the rest of eternity taking smaller and smaller tiles -- until the approximation becomes so good that for all intents and purposes it is the area -- is known as THE INTEGRAL.
Here, then, is the defining beast of integral calculus. You now have an incredibly convoluted way to find the area of a circle (hooray!). Well, silly, you can probably see where this is headed. The same game can be applied to find the area of any shape we like. Shapes that don't have a handy formula for their area. The shadow of a daffodil, for example. Or a brontosaurs. The area of Australia or the island nation of Great Britain. A coaster in the shape of Al Gore. Whatever. As long as we can describe the shape in numbers (in math lingo: we have an equation that defines the outline of the shape) we can form an integral and find the area.
And remember - the answer is not just an approximation. It's the real deal. The true value of the area. The end result of taking more and more smaller and smaller tiles, forever, unbelievable though that may be.
Here's the notation used for an integral: ∫. It's supposed to remind you of a rounded Greek letter sigma (∑), which is the universal nerd symbol for "add up stuff." There's a little more to the notation, and we give the interested a few more details below. But you now know the thrust, the gist, the meat of integral calculus. Crazy!
And so here we pat the integral on the head and send it on its way. Next up: differential calculus. However, like a bad penny, like Gollum, like a certain psycho ex-girlfriend, the integral is going to turn up again before our tale is finished. For it turns out there is more to the integral than just finding areas.
We may be through the the integral, but the integral is not through with us (dun, dun, dunn....)
Here we provide a few integral bonus points-of-interest. These tips will help you begin to understand calculus information you may see elsewhere, either in textbooks or on-line. In fact, spend a few moments digesting this info then pop on over to a Wikipedia calculus page (we provide a link below) and see if you don't start seeing some familiar faces in what may have previously been an indecipherable jungle of Greek symbols.
Bonus Knowledge #1: Calculus jocks call the process of repeating a calculation over and over using smaller and smaller thingys (tiles, squares, whatever): "taking a limit." This is usually abbreviated "lim" and the abbreviation usually includes an indication of what is being made small. So, for example: lim Δx → 0 indicates the limit as Δx is squished to zero. Oops, we have snuck in another notation: in an integral, Δ is used to indicate "the typical size of...". In some expressions (see next tip) we abbreviate all this limit notation argle-bargle by simply writing "dx" instead of "lim Δx → 0". Think of this as calculus shouting: "HEY YOU! DON"T FORGET THIS IS THE RESULT OF TAKING A LIMIT."
Bonus Knowledge #2: A bona fide integral uses a little more notation than we have let on. Here's how a calculus jock would do the circle example: First, draw an x-axis through the circle and throw out the bottom half (see accompanying figure). We're going to find the area of the top half and multiply by two. Why? Because it makes things easier if one of the boundaries of our shape is the x-axis. Second, instead of a tiling of squares, we use a tiling of tall-and-skinny rectangles. Pick a rectangle width -- call it Δx -- and make up a name for the height of each rectangle (where it crosses edge of the semi-circle). We'll use the symbol f(x) to indicate the height of the circle above the place x. The area of a rectangle centered at x is f(x)Δx and by summing (∑) all rectangles we get one tiling approximation. As described in Bonus Knowledge #1, the true value of the area -- the integral -- is the limit of these sums as Δx → 0. That integral is properly written as ∫ f(x) dx. The f(x) indicates what you're taking the area under. The dx indicates the direction we're summing along -- the rounded "d" a reminder that the integral is the end-result of a limit applied to the Δx. We usually add little numbers to the bottom and top of the integral sign to indicate where we start and stop the summing process. In words, one says "the integral of f(x) from 0 to 1."
Bonus Knowledge #3: As soon as mathematicians worked out areas, they went bananas applying the integral to other types of objects. Hopefully you can convince yourself that by using cubes instead of tiles, we could play a similar game and find the volume of a sphere. Or the volume of anything. Einstein introduced four dimensional objects -- and modern physics goes up to 11 -- and mathematicians were soon using integrals to study so-called "hyper-volumes" that you can't even draw. It sounds scary, but the concept is the same. We can also go to fewer dimensions. That is, we can find the length of a curly string by using a collection of little straight line segments. The length of the string is equal to the number of line segments times the length of one segment. As we use more and more of shorter and shorter segments, our collection of straight lines becomes a better and better approximation to the true bendy path of the string, No matter what we're integrating, an integral formula will contain some indication as to what's being summed. You'll commonly see: dx (as we have done in our example above), dA (equivalent to: f(x) dx -- a little tile of area), dV (a little cube of volume), or dS (a little piece of string length).
Achievement Unlocked: Here's the wikipedia page on the integral. You should be able to go there and recognize the notation and perhaps the description. Don't beat yourself up if there's stuff there you don't understand (c'mon you've only been studying calculus for, what, 20 minutes?)
Calculus on Amazon
Here is a biography of Alexander. Nowhere -- on not a single page -- will you find anything about calculus. That's right: you now know more calculus than Alexander. Not so Great after all, was he?
Next time on What Is Calculus?
If Peter Jackson has taught us anything, it's why make one movie when a trilogy gets you triple the box. Oh, we're just funnin' ya, Mr. Jackson -- what you really taught us was that bit with Gandalf bringing the company to Beorn one-by-one rather than risk rebuke by dumping a big dwarf-pile on the stoop all at once.
More to the point, we have reached a convenient stopping place in our tale. Or we should say pausing place, for to come this far and not read the triumphant final chapters of the calculus story would be like flying all the way from Jersey to Naples and not going to see the volcano because you spent all week in the hotel room whacked-out on heroin (you know, it occurs to us that much of our conversation is based upon obscure movie and TV references, which may begin to explain the confused looks we encounter in everyday life).
So reward yourself for coming this far, perhaps with a fine LabKitty beverage-related product. Rest and digest, repose and reflect upon your budding calculus intimacy.
And when you're ready for more, click HERE. For next time on WHAT IS CALCULUS?, we meet Calculus' better half: the derivative!
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