# what does the word "Integrate" mean?

Discussion in 'Off Topic [BG]' started by AmazingGracePlayer, Dec 10, 2006.

1. Well, mathematically, what does it mean when someone says "integrate the following"

I looked in my math book, dictionary.com, and meriamwebster.com and the definition was "to find the integral."

I don't know if I should find the definite integral (Derivative) or the indefinite integral (anti-derivative)?

2. ### Neb MaroI don't think, but I still am.

Oct 20, 2006
So. Cali
I thought integrate meant to mix or to bring together.

Sorry, didn't see "mathematically" the first time I read your post.
Tried looking for the definition of integral, but couldn't connect it to integrate.

3. in maths integrating is the the opposite of differentiating (sp?)

4. ### Neb MaroI don't think, but I still am.

Oct 20, 2006
So. Cali
So to define integrating mathematically speaking, you need to first define differentiating?

5. Give me the problem and I'll solve it and explain, if you like.

6. nah, i cant be bothered defining it, just telling him what to do

7. Well this is where you're wrong. The derivative operator is not the same thing as a definite integral.

Let's define the derivative (d/dx) as the operator D(x), and the integral wrt x as the operator I(x). This is actually notation you will see at some point, but I'm just using it because it's easier on a forum.

The integral operator is just the reverse of the derivative operator: D(I(x)) = x.

A definite integral is simply the integral across two values of a function - say I(x) |[SUB]a[/SUB][SUP]b[/SUP] = b - a
An indefinite integral is the general antiderivative of a function - eg. I(x) = x[SUP]2[/SUP] + C where C is any constant.

When someone asks you to integrate a function, unless they give a domain on which to integrate they will generally mean to take the indefinite integral. The constant factor is important at the early stages of this concept, but as you will later see there are many techniques for dealing with it.

8. if the problem gives you limits of integration, either explicitly or by the context of the problem ("find the area," etc.), then it's a definite integral.

if you're just given a function and it says "integrate!", then you are more than likely just asked to perform indefinite integration. and don't forget your "+c"!

9. Yup. The integral symbol is this: Which is just a big "s" standing for "sum". I'm serious.

Basically, if you have a curve like this:

And you want to find the area under the curve from 0 to 3 (doesn't matter), you have to integrate.

How would you normally do it? Have you learned about Riemann sums? Where you divide the curve into rectangles and add them up?

That's a Riemann sum. Now, you can obviously see that it's not perfect. Some parts stick out, some parts don't go all the way. So what Calculus does is make the rectangles infinately small and then you add up infinately many of them. No error.

So that's what an integral does. It's the area under the curve of a function. If you have a function that says where you are at a given time, then you can take the integral of it and see how far you've traveled. Things like that.

How to do it is a lot harder. This is where anti-derivatives come in. You take the anti-derivative of a function, then evaluate it at its end points.

So for the parabola, the function is x^2, right? If you wanted to find the area under the curve from 0 to 3, you'd take the anti-derivative, (1/3)x^3, and plug in 3 and 0, so

(1/3)3^3 - (1/3)0^3 = 9. That's the area under that curve.

10. Ah, the usual bunch of us physics and engineering students come to save the day. Now all we need is Geoff and we have a party!

11. w00t math party! I'll bring the calculators!

Actually, I just upgraded from a ti-83 plus to a ti-89 titanium yesterday. What a huge jump! Well, I don't use it much, but I don't like integrating stuff so I'll have my calculator do it. Especially in Fourier series, when the integral isn't the main part of the problem, just a step. I don't want to waste time on it. 12. Yeah, but when you're at home you can use matlab/maple/mathCAD... and on the exams, usually there's no Calculator allowed anyway! 13. ti-89 ftw! 14. To make it really simple:

The indefinite integral is that function which, when differentiated, returns the function you are integrating.

The integral of X would be .5X^2, because the derivate of X^2 is 2X... and when you multiply by .5, you get just X. With a little practice and common sense, you would be able to integrate all of the basic functions you can differentiate.

It gets a little troublesome once you start dealing with ln(x), e^f(x), trig functions etc... but you'll survive. Every special function has its own method of integration that's just waiting for you to memorize!

15. Actually, we can use calculators on the exam. That's why I bought it. At home I just use this:

http://integrals.wolfram.com/index.jsp

I have Mathematica, but it's such a pain to use. Here they already set it up and ready to go. Don't have to wait for it to load up, either.

16. im running this one

17. Party it is. The bases look pretty much covered. The only thing I have to add is that a Jeppesen Flight Computer is a terrible device. The thing is basically a slide rule (though it's a circle) that you can do multiplication and division on one side, and cosine and sine on the other. Then people think of really terrible problems to solve with it, but I digress.

18. Lucky you - we don't. Not in math classes anyway...

Cool link - thanks Hey now, slide rules are great! Now the JFC, after reading about it briefly, I'll agree is junk 19. What math are you taking? It's assumed in my class that you already know how to integrate, so you shouldn't be tested on that, only on the things we covered.

Sure, I can integrate xe^(-ix&#920 , but that takes like a quarter of a page and there's a lot of room to screw up, and it's time consuming.

20. You've definitely got the "how to integrate" thing covered...

As to the original question "what does the word integrate mean?" It means "find the area under a curve."