I REALLY can't believe it's not area

[K.I.L.E.R]

Integration is about adding things.
This actually explains a whole load of things such as what I've been reading about integrals.
Normally they are used to compute area but I've seen them used everywhere. People mentioned that your adding these terms and all I see is an integral sign.
It weirded me out like a soup on a butterfly.

My teacher today explained what integration was really about and I was like "*Shocked*".
I have a whole new perspective on things, in fact I've thought back about certain things I had trouble understand and are now crystal clear.

Who'd have thought that such a simple piece of information could lead to crystalising your understanding in vast areas of study?

 

[KimB]

Well, integration is area, but it's also very important to understand that it's also a set of discrete additions taken to the continuum limit.

One way of writing down an integral of f(x) from a to b is, for example:

lim_(N->infinity) [sum(i = 0 to N, f(a + idx)*dx]

where dx = (b-a)/N.

In the above integral, when you take i = 0, the you will have f(a) within the sum. When i = N, you will have f(b) within the sum. So what you're doing is summing up the areas of N different sections of the integral, starting with x=a and ending at x=b. Take the limit as N->infinity, and you have the whole area under the curve between a and b.

But it is worth noting that this is the area under a curve. It can be a bit confusing when the area under a particular curve has some other physical meaning (like, say, length). In these situations it's good to fall back on the "taking many sums" idea of what an integral is.

 

[K.I.L.E.R]

My teacher explained that taking an area is taking a sum and an integral does that. The alternative is to type up 100000 or so values and do it by calculator or hand.
So it's not really an idea, just the purpose of integrals.

In these situations it's good to fall back on the "taking many sums" idea of what an integral is.

 

[KimB]

Well, these are all conceptual ideas. Integrals can be defined in a number of different ways. The important thing is being able to understand where, how, and why they're used. You won't in practice ever divide up an integral into N parts and sum them (it's horribly inefficient). But it's a very useful conceptual tool to understand what's going on, for most applications.

 

[K.I.L.E.R]

So really it is just another way of doing addition but conceptually it can be thought up in different ways to better someone's understanding?

Sorry this is a little confusing now.

 

[KimB]

Well, right. Sometimes it's useful, for example, to look at the definition of integration as the inverse of derivation. For example:

integral(dy/dx dx) from a to b = y(b) - y(a)

You're still just taking the sum of the function dy/dx evaluated at many points along the integral, but in this case it can be better to think of it as the inverse of the derivative. The trick is taking the definition of the integral which most easily gets you to conceptual understanding of the problem at hand.

 

[K.I.L.E.R]

I'd be lost without you.
Thanks.