Concepts vs Formulas
Here we go with another nerdy post, because how much more nerdy can I be?
It's near the finals and I am sighing in front of my computer as I type this sentence out. Heh, that one last thing you must do before the semester ends...reading, revising, cramming, exercising, stressing the hell out of your brain trying to do as good as possible in it for the sake of scholarships. So today's calculus class is a little different - we are using it as a revision class instead. It's a revision class on some topics on integration including the fundamental theorem of calculus, calculating area, calculating volume, and stuff like that. In general, things that we have already done...and forgot. Well, we have so many topics rolling in day by day and it's really hard to memorize every single thing...wew.
In all those questions, we had something like this.
Find the volume of a tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths 3 cm, 4 cm and 5 cm
The first thing that came onto my mind - what the hell is a tetrahedron?
It's not only I have no idea how one looks like, all those mutually perpendicular stuff are blowing my mind. What on earth...is this calculus or geometry class? Google image searching does not really help because all we see is how a tetrahedron looks like, with...no information on what all those mutually perpendicular thingy actually mean.
Whatever, we peeked at out lecturer's answer sheet and found out that it is something like this.
Uh, forgive my tablet drawing, it is just too hard to use a ruler with this thing...and there is no such thing as a ruler tool in Microsoft Edge. But at least that gave me a good idea on how the shape actually looks like, so it is possible for me to finally find it's volume then.
To find the volume using some integration method, the generic way is to find out a formula that you can integrate that results in the entire volume. The correct method here is to integrate the formula of the area of one horizontal triangle from 0 to 5 along the y-axis, and the lecturer used this method too. But, this method just didn't come into my mind this morning - well, it's morning so I cannot expect my mind to fully function anyways, but some other method came into my mind instead,
When you say about integration for volume, it is about slicing the entire shape into infinitely many almost-flat slices that you know how to find their volume, and finally sum them up. That is called the Reimann sum, and is somehow the fundamental method of integration. It is quite an important topic in exams because the lecturer assured us that she will definitely put one question on it with heavy mark allocation in the finals, just because of how important it is and how it shows the entire concept of integration.
Here for this shape, all I can see at that time is, I can slice it horizontally into infinitely many pieces, have their volumes as x * (3/4)x * 1/2 multiplied by delta-y, and hence do the sigma stuff to get my solution. At that point I thought that I could change the sum equation into something I can directly integrate, but all I have is just xs with a delta-y behind there...it just does not work that way.
What to do? Summing them up will still work, right?
So, I indeed summed up my idea which is impossible to write in a standard integral formula, in a Reimann sum fashion. And the answer is correct.
When I asked my lecturer for verification, she said that she never saw anything like this before - and in fact, it is correct. The issue with my method is because I have my height and area in different terms (x and y), so it is almost impossible to do direct integration like most people do. She then showed me her solution, which is significantly shorter than mine, and I can easily convert my solution into hers by building another formula from the diagram and substitute the Xs and Ys in correctly.
This is something I consider very beautiful in mathematics - you can have many methods to reach one single solution, and all of them will not contradict each other - everything just fits in magically. There's a joke saying that mathematics is actually a kind of religion, and in some ways I think I can agree because it is just a little too hard to explain why everything can fall into place automatically and almost magically.
If I remembered the formula and not the concept, I wouldn't be able to solve this. "If you can't think of anything else in exam, just write whatever you can think of at that time - like this Reimann Sum thingy, it might take a longer time but it's better than leaving it blank or just wasting more time to think of another probably wrong solution". That's what my lecturer said, and to be honest, it is impossible to do so without an understanding of the concept behind the magic we call as integration.
At any time, I will still say that having the correct concepts is more important than memorizing formulas. You could have done good in exams with formulas, but when the question twists or you are just unlucky enough to not think of it correctly at that time, the concepts can actually help and remind you of the correct method to interpret the subject and help you to tinker out a method to be used, right on the spot. Plus, solving real life questions rely more on concepts - well, I don't think we can find any real life applications that need you to use calculus in real life anyways...but that's a thing I guess?
Lesson of story: Always understand stuff behind formulas. If it happens that you are unlucky to get a lecturer that can't explain...use YouTube. 3Blue1Brown is a super good channel, trust me.
Ah right, I guess I have been studying a little too much and is too stressed till the fact that I wrote this thing out...see you next time anyways
--Lilacse