by nsguy1350 » Tue Jun 12, 2012 12:33 am
Integration is used for finding areas under curves. There are two (basic) kinds of integrals: definite integrals, and indefinite integrals.
The symbol for both is the same, except in one aspect: definite integrals have upper and lower "limits", or points of integration. On the number sense test, you will only see definite integrals. Definite integrals are used for finding areas, and require indefinite integration (or a numerical approximation. That's how your calculator integrates, unless it has a computer algebra system that can do definite integration the "right" way). Indefinite integration deals with finding antiderivatives. An example of an indefinite integral is [unparseable or potentially dangerous latex formula], and an example of a definite integral is [unparseable or potentially dangerous latex formula].
Just as it sounds, an antiderivative is basically the opposite of a derivative. And so, [unparseable or potentially dangerous latex formula]. (The dx is used to say you're integrating with respect to x, but don't worry about it here. It can be thought of as the derivative of x for now.)
But wait; [unparseable or potentially dangerous latex formula]. If you add a constant to the function, and take the derivative, you get the same derivative! So which one should you pick when you find an antiderivative? Surely, we do not say [unparseable or potentially dangerous latex formula] (same derivative as the other two). You have one choice out of an infinite number! Because of this, we indicate the antiderivative of a function [unparseable or potentially dangerous latex formula] as [unparseable or potentially dangerous latex formula], where C is some arbitrary constant. Therefore, [unparseable or potentially dangerous latex formula].
Well that's indefinite integrals (indefinite = finding antiderivatives) for you. Here's definite integration.
Definite integration is where you find areas under curves. An example of a definite integral is [unparseable or potentially dangerous latex formula]. Here, you are "integrating 2x from 0 to 2," or finding the area under 2x from 0 to 2. 0 is your lower limit here, while 2 is your upper limit. First, geometrically, find the area under 2x (it's a triangle. Since this is in the NS forum, do it in your head without using pen, calculator, or anything besides your brain :P) from 0 to 2 (oh yea, one more thing: area means from the x-axis to the function. If a function is negative, it means "negative area" from the function to the x-axis. Yes, in Calculus, you can have "negative area", in a sense. No area is negative, but we can treat it as negative when doing things such as integrating velocity functions and such.)
Alright, well hopefully you got that. Now, the actual process of definite integration.
To integrate a function from a lower limit to an upper limit, we use the Fundamental Theorem of Calculus (FTC). There are actually two parts; the one we are going to talk about deals with finding areas using antiderivatives. The other part, basically, deals with how differentiation/antidifferentiation are opposite of each other, and how doing one to the other reverses the whole thing --- find the derivative of the antiderivative of [unparseable or potentially dangerous latex formula] to see what I mean, for example.
Alright, so basically, [unparseable or potentially dangerous latex formula]. (Ahh, I used \displaystyle and a space, looks better now :D) That's one part of the FTC for you.
So, going back to our [unparseable or potentially dangerous latex formula], we see that [unparseable or potentially dangerous latex formula]. Since we evaluate the antiderivative at two points when we integrate, we'll be using that antiderivative. Thus, [unparseable or potentially dangerous latex formula], where | indicates evaluation. This simplifies to [unparseable or potentially dangerous latex formula] As you can see, the answer 4 is what you got geometrically (I would hope). Also, do you see how the constants cancel? This means that when finding the area under a curve, do not worry about constants at all; they will always cancel. Constants of integration are only for indefinite integration. Thus, we can write that a bit more concisely:
[unparseable or potentially dangerous latex formula]
Also, taking derivatives is easy. Taking antiderivatives is a bit harder, and some functions are either REALLY hard or impossible (as far as we know) to integrate, at least using elementary functions (go look up on integration if you wish to know more.) Basically, you reverse differentiation. Get used to doing both, and then you'll become fit to do them during the NS test. I won't go too far into rules of integration, as you can just reverse the differentiation ones (remember, this is for the basics, there are many more advanced techniques!)
So yea, I won't go too far into rules and things. If there is something you need clearance on, ask, or if there is a mistake, please tell me.
For practice: antidifferentiate [unparseable or potentially dangerous latex formula] (Hint: Reverse the chain rule!). Then, find the area under each from 1 to 3.
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