It depends on what level you're doing. If you've just been told that "integration is the opposite of differentiation", and that it has something to do with infinitely thin strips, then it can be difficult to see _why_ things are happening.

The fundamental law of calculus does pretty much state that integration is the inverse of differentiation, ie that:

dF(x)
if ----- = f(x)
dx

then F(x) is *an* antiderivative of f(x) with respect to x. I say "an," because there are many antiderivatives of a function in general.

OK, that's all well and good but it doesn't help solve integrals. It may help get an understanding of them and to understand some proofs, but when it comes to solving, you'll need to do something.

First off, remember the above statement. Simple integrals can be done in your head if you can think of an antiderivative. Take f(x) = 2x, for example. F(x) = x^2 is an antiderivative just by inspection. Even some more complicated functions (trigonometric or exponential) can be solved this way. But the bad news is that there is no foolproof method to solving all integrals. Instead, you'll just have to learn or derive certain tricks, like substitution.

Integration can be thought of using the strips argument, that being that given the function below:

f(x) ^
| ___
| / \
| / \_
|\__/
|
|
+------------> x
a | b

The integral of f(x) with respect to x between the limits a and b is the area bounded by f(x), x=a, x=b, and the x-axis. To do this, split the interval [a,b] into strips. The first strip is marked as |, and is at x=(a+h). The area under the function between a and a+h is approximately a parallelogram with area (h) * (f(a) + f(a+h)) / 2. The principle of integration is that as h tends to 0, the strips tend to completely fill the area without leaving any "corners." So the limit of the sum of all of the strips as h tends to 0 is the integral over the interval [a,b].

That was a rather informal and very short thing, which does take a lot for granted. For one, it doesn't discuss the idea of continuity or even the existence of a limit, so don't get quoting me on it. The best advice I can give is to get a Calculus book from your school (or local) library. Chances are, your tutor will be able to recommend a good one. If you're still able to, talk to your tutor about it. They've got qualifications in that sort of thing, and hopefully they'll be able to discuss things in more depth with you face-to-face, or at least face-to-class.

Mat"Please say I got that right. I've just finished my first year of university about it..."thew