Get out of my head, please. I don't know why I like you so much.
Just because someone likes the same bizarro crap that you do doesn't make them your soulmate.
Right? Right.

Now, I'm going to do that crazy calculus spewing blog I usually do before I test. I think it helps, just to get all my knowledge (or lack thereof) in one place. Usually this is on tumblr, but I'm banned from tumblr until finals are over, so here goes!
INTEGRALS:
Are essentially the opposites of derivatives. They usually have a little dx at the end and you are finding an indefinite one, you MUST add +C at the end or you will get points taken off. Generally, using the reverse power rules works pretty well, when you get x to the n+1/ n+1. And if you have any coefficients, take them outside of the integral.
Note to self: study up on the fundamental theorum of calculus.. jsut because
then there's u substitution. You take the more complicated expression and call it u. Take the deriv an call that du, so that du equals the part of the expression that isn't u so that you can cancel it out and replace all the parts of the expression with either u or du. Yep.
Sometimes, though, if you can't cancel it out, you did something wrong.. Either you picked the wrong thing to be your u, then you have to try again. Or it jsut won't work so you have to add an extra step and solve for x so that you can plug in u into the expression.
I think that's all there is to it for integrals. Oh, and the area under the curve is the integral of the function at the given interval.

I don't know how to do limit summation, so I will be back tomorrow with that wonderfully classified information