How to make the most out of Year 11

On my Stats page, it says that someone got onto this blog by Googling, "how to make the most of year 11," which sparked off some inspiration for this blog post. I'm a pretty poor person to be giving advice like this, since I didn't exactly have the best time in year 11 (I did well academically but not in much else), so this list is mainly a mix of things that I did do well coupled with some things that I wish that I knew. Of course, as I just said, my year 11 wasn't exactly perfect, so don't rely totally on my advice.

Studying

• Try and live and experience the subject as though it's a major part of your life or something. Then you might remember it better. It's a bit hard to really explain what I mean here- I guess the thing is to get really mentally and perhaps emotionally involved in your studies so that your new knowledge becomes a part of you. (Note: A friend just pointed out to me how unhealthy this sounds. He's probably right- I wasn't exactly the most emotionally healthy person last year. I think that perhaps the most important part would be not to see the stuff you learn at school as just dry chunks of information to memorise for tests, but rather to see it as interesting and maybe somewhat applicable to your life as well.)
• When learning new stuff, try and connect it to stuff that you already know, or other stuff that you're learning at the same time. More connections makes the material stick better.
• Don't be afraid to try unconventional methods of studying. Writing this blog has, for me, been a useful tool to help me studying. Also try using multiple different methods that you can use at different times depending on your mood.
• Study timetables work for some people (though admittedly I've never really used one), but if you have a really busy day, be prepared to be at least a little bit flexible with yours so that you don't get bogged down with guilt.
• If you're pretty much solid on whatever it is that your class is covering now, look at the course outline and see what you're meant to be doing next so that you can be ahead of the class.
• Try not to just read passively- at least take notes or do something other than just reading and mindlessly highlighting. One way I keep myself focused on what I'm reading is to keep a pencil handy so that I can write stupid comments in the margin related to whatever it is that I'm reading (hey, you have to understand at least a little bit of what you're reading to comment on it!). (This is another way I used to keep myself from over-stressing in tests and exams, but I've been advised not to do so because apparently the markers don't like it, so it's back to the drawing board I guess.)
• Here's a brief guide as to how I studied for all my subjects last year: Lit- writing practise essays and giving them to the teachers for feedback, Maths/Spec/Chem- doing every single practise question that I could get your hands on, Chinese- I was probably a bit unbalanced here but I mainly just did Language Perfect and read Chinese books, though I did also visit China twice last year on two two-week trips, Music- doing every aural exercise that I was given (though possibly could have done more on the website that my school signed up to- http://www.e-lr.com.au/), writing blog posts on the Lit parts, reading all the given handouts and relevant textbook sections, listening to the set works and practising clarinet. I didn't do much theory study because that's probably my strongest section, but one thing that I've done this year is look up all the Italian terms that I don't know on that list of terms that they gave us, and add them into Anki, a free flashcard program that you can download at http://ankisrs.net/.
• Here are some ways that you could study outside the course and feel extra confident: Lit- read other classics, particularly those written around the time or in the same genre as your set texts (for example, if you're studying Rosencrantz and Guildenstern, an Absurdist play, you could read Waiting for Godot, another Absurdist play written about 10 years earlier or so), or read scholarly articles on your set texts or other books that you've read (use Google to find these articles), Maths/Spec/Chem- find other textbooks to get alternative methods of doing stuff you already know or learn new things, use the Internet or library books to find out more about the topics that you're studying, Chinese- read books, watch movies, talk to people, practise writing sentences (you could start doing this by picking some sentences in a book you're reading and then changing some words so that you get the feel of different sentence structures), get a penpal and write to them (I haven't written to mine in a while...), Music- learn more challenging repertoire, learn the basics of another instrument, write compositions, read music theory textbooks, read about the music, art and culture of whatever time periods you're studying.
• Obviously, be a nice person and help your classmates if they need it, because you will also benefit by explaining stuff to them. The more you have to put your thoughts into words, the better you will come to understand the topic.

Work/Life Balance

Hm. I'm not very good at this part. All I can really say is, unless you have an assignment due the next day, pick a bedtime and a time to stop working (maybe about 20-30min or so before bedtime) and stick with it. Don't stay up super late studying the night before a test because you'll probably do better with less study and more sleep than the other way around.

Oh and also make sure that you've got SOMETHING that you do besides studying. Again, I was a study-holic in year 11 (still am in some ways, but I'm too burnt out now to do as much studying as I did last year), but I at least had a few things that I did to break it up. I did driving lessons and played clarinet in band. This year, I'm in my school's wind quintet, orchestra and wind band, and I also do yoga since my year 12 year coordinator encouraged/forced me to do it and it's actually pretty fun. I no longer do driving lessons since I've passed my test (albeit only on automatic, so I can't drive Aston Martins since I'm pretty sure most if not all of them are manuals), but I still go driving on weekends to build up my hours, not like I'm in any hurry to do so though.

I'll probably add to this page later. For now, enjoy year 11 or year 12 or whatever year you happen to be in! I'm probably not going to heed my own advice on that one, but... whatever. To quote the toilet wall at my school:

"Have a nice day- you're lucky to be alive, appreciate it."

"NO THANKS"

My second take on the Volumes of Revolution formula

Another way of looking at the Volumes of Revolution formula is to divide the area up into a series of... cylinders. Yes. You know how when you were learning about how integration "worked" you had those investigations where you had to divide the area up into small rectangles and calculate the combined area and work out an estimate? Well, that's exactly what I'm doing here, except with rectangles rotated 360 degrees about one of the axes because that just totally makes them less obnoxious. (Not really.)

Here's a triangle that wants to become a cone.

Now, one way that you could work out the volume of the cone-to-be is by using the area-of-base-times-height-divided-by-3 formula. But let's make stuff more complicated, because that's so much fun! (Okay, I should probably stop with the sarcasm now.)

One place to start off would be to divide the area between x = 0 and x = 2 into four rectangles and calculate the volume of each corresponding cylinder and then add the volumes together. (The fourth rectangle in the diagram below is that purple line at the bottom that has 0 area. I really should have picked (y = x^2 +3) or something else that would have given me a better area- an actual area, in fact- for the first rectangle. Ah well.)

As can be seen from the above diagram, these rectangles don't give a very good approximation of the area. Look at all those triangle bits being left out! To counter this partially, you have to divide the area into smaller rectangles like so:

And so on, until the rectangles are all so small that all you see is this vague purple blur.

It's easier to work out the area of each cylinder (formed from each rectangle) if you make yourself a nice simple formula and then work from there. The volume of each cylinder is (pi)(r^2)(h), but that's pretty generic so let's look at how it applies to the above diagram.

For the above diagram with the 8 rectangles (or rather, 7 rectangles plus one pathetic excuse of a rectangle), each cylinder is 2/8 = 1/4 units wide. Therefore, h in this case is 1/4 units for every rectangle. r is the tricky one, because it is different for each rectangle. r is equal to the y value at each x-coordinate. For the first rectangle, r = 0, then for the second one r = 1/4, then for the third rectangle r = 1/2, and so on (it sure helps having y = x as your formula! If it was x^2, though, then the rectangles' r values would be 0, then 1/16 and then 1/4 and so on.) Finally, pi is a constant, which is nice.

Now, as I said before, you eventually want to get a whole load of infinitesimally small rectangles, and therefore infinitesimally small cylinders too. Let's say that you want n small rectangles in that space. Therefore, the width of each rectangle is 1/n units. That gives us (1/n)(pi)(r^2). Now we need to work out a general formula for r^2. For the first rectangle, r = f(0), then, in the second rectangle, r = f(1/n), then r = f (2/n) and so on. A general formula for r would be f((p-1)(1/n)), or f((p-1)/n), where p is what number rectangle you're looking at.

To sum up:

Volume of first cylinder = (1/n)(pi)(f(0/n)^2)
Volume of second cylinder: (1/n)(pi)(f(1/n)^2)
Volume of final (nth) cylinder: (1/n)(pi)(f((n-1)/n)^2)

Combined volume of cylinders: (1/n)(pi)(f(0/n)^2 + f(1/n)^2 + f(2/n)^2 ... f((n-1)/n)^2)

Which can further be summed up in the handy dandy summation formula that I just learned! (Too late for the previous investigation, unfortunately, but hey, better late than never!)

Now, to find the exact area under the curve, you want n to be as large as possible- infinitely large, in fact. That's where limits come in! Now, I'm not sure whether I've used limits correctly in the following equation, so as soon as I've published this I'll shoot off an email to my teachers to ask them. Of course, if you're a teacher yourself, or if you're just plain good at maths, then you can tell me whether I'm right or wrong in my notation.

Let's analyse this formula a bit more. f((p-1)/n) gives the y-value for various x-coordinates, since f at something gives the y-value of whatever x-coordinate "something" is. (f((p-1)/n))^2 is therefore where the y^2 part from the volumes of revolution formula comes from. Pi, our good friend, is there. Finally, the 1/n is the width of the tiny little rectangles/cylinders, made even tinier as n approaches infinity. (1/(a large number) = a very small number.) I believe this is where the "dx" part comes from in the volumes of revolution formula. The volumes of revolution formula also has the start and end point marked in, but I'm not sure how to include that in the summation formula above.

And that is a second explanation for the Volumes of Revolution formula, just because one explanation is simply not enough!

My take on the Volumes of Revolution formula

Okay, well, being the nerd I am I feel like explaining how I remember the Volumes of Revolution formula.

Volumes of revolution, if you didn't know, is basically the volume of the solid you get when you rotate some area of a graph 360 degrees around either the x-axis or y-axis. For example, in the diagram below, when you rotate the shaded bit between the lines x = 0, y = 0, x = 1 and y = 2 about the y-axis (or x-axis for that matter), you get a cylinder. If you don't see how this works, then tape some paper onto a pencil and twirl it around a bit.

Here are the two formulae for working out the volume of the solid formed. The top one is for rotating around the x-axis, the bottom one for the y-axis. In the first equation, a and b are the x-coordinates of the "start" and "end" points of the volume to be calculated, while in the second equation, a and b are the y-coordinates of the "start" and "end" points.

You might be thinking, but wait! How do I integrate with respect to x when I only have ys there, and vice versa? Well, you just have to rearrange the equation. For example, if your equation is y = x^2, and you want to find the volume when you rotate an area under that graph around the x-axis, then you have to rearrange it to get y^2 = (x^2)^2 = x^4, and then substitute x^4 into the first equation. Then all you have to do is solve it like a definite integral.

Now, here's the simple(r) and probably very mathematically incorrect explanation as to why this formula works.

I'm going to put the above graph here again, just so you don't have to scroll back up just to see what I'm blabbing on about.

As you can see, this to-be cylinder either has a height of 2 units and a radius of 1, or vice versa, depending on whether you're going to be rotating the shaded bit about the y-axis or x-axis. For now, we're going to rotate it around the y-axis, because I tend to think of cylinders as being long thin things, because my thinking totally conforms to stereotypes like that. To find the area of this cylinder, then, you could just use our good ol' formula back from year 8 or maybe from primary school if you were lucky enough to have teachers willing to teach that stuff there: the good ol' (pi)(r^2)(h) formula for finding out the volume of cylinders.

Now wait! In the above diagram, since we're rotating about the y-axis, the radius is equal to x. That's how we get the (pi)x^2 part of the formula. Now, as for y, and the whole dy and integral and everything else that's in the volume of integration formula, we have to think outside the orange rectangular box above and start thinking about all kinds of weird irregular shapes.

The (pi)(x^2) part gives the area of any given little slither of area rotated around the y-axis. To actually turn that into volume, we have to multiply that by height as well. We could just multiply by the height of the entire area in the above diagram, but remember: most curves do all kinds of weird things. The radius, or x^2, generally changes as you go along the curve, so you have to multiply by the height of infinitesimally small sections at a time to get the volume. Here's a weird curvy thing that I drew to explain this better:

Note that at y = 2, x, and therefore x^2, is much bigger than it is at x = 1. Every little section along this weird curvy thing has a different x-value too. Therefore, to calculate the total volume of this curve between y = 1 and y = 2, we have to calculate the volume of every tiny little slither of volume between these two points and add them all together.

Thankfully, someone smart invented integration, so we don't have to do all that. I think that the "dy" part of the integration formula is a reference to all the small slithers, a bit like how (delta)y is used to symbolise a small change in y.

Therefore, to recap: the (pi)(x^2) in the integration formula is the circular area of one of the little bits of area under the curve rotated around the y-axis, while the dy is the height of each slither. Then the b and a tell you where to start and stop adding up slithers. And then that fancy integration sign is there because it's fun to draw and makes you feel like a maths genius every time you do so.

For the formula for rotating around the x-axis, it works much the same way: the (pi)(y^2) in the integration formula is the circular area of one of the little bits of area under the curve rotated around the x-axis, while the dx is the width of each slither.

Gosh, I could almost be talking about snakes with the many times that I've said the word "slithers!"