I'm just curious if someone could give a a quick (or not-so-quick, whatever) rundown on integrals. I.e. what they are, what the notation is, what they stand for, etc.
I understand derivatives but I'm trying to see where this whole integral thing is going.

Yes I surmised that much. But I mean what is unique to integrals? Do they have their own sets of rules? What do they mean (i.e. like differentials are the instantaneous rate of change), what are the different types of notation, etc.?

They can also be used for calculating various other sums and other hardcore shit. Volumes for example, or the sum of some quantity along a path (say, you cycle through a cloud of mosquitoes. If you know how they are distributed in the cloud and know your path through the cloud, you can integrate the sum of how many of them you'll catch in your mouth). I have very little idea how, just heard the big boys talk about it one day. Areas are definitely the most common and intuitive application though, just showing off here.

Integrals are essentially used to get areas under curves.

If you've done Reimann sums yet, you know that you can cut sections of the curve into a bunch of rectangles, multiply the bases (Delta X) together with the heights (Y values), and approximate the area under the curve. Well with integrals, you do the same thing, except the you take the limit of Delta X as it approaches zero (turning it into dx: a very, very small distance). This time though, the area under the curve is (practically) exact.

There are three parts to the basic integral. That fatass "S" thing in front is known as Sigma, and basically tells you to add. Following that, there's the f(x) equation (the "height" of the rectangles). Finally, there's dx (the infinitesmally small "width" of the rectangles). So the Sigma up in front tells you to add all the little areas of dx*f(x) together.

It might sound convoluted at first, but it'll eventually click. You're going to really love calc II next year, when they try to tell you that 1 the the infinite power is fricking 2.718. Now THAT'S convoluted. (1x1x1x1x1x1... = ...2?)

As for integral applications, you'll eventually use them to figure out how long curves actually are, or to find a path equation for an electron in particle physics. Pretty much every physical science field uses integrals in some way or another. They're pretty integral (AHAHAHAHAHAha...ha...haah...) to physics jobs.