It's been a while for me, but isn't what you just said the first fundamental theorem of calculus? The second is:
If f is a continuous real-valued function on some closed interval [a, b] with the anti-derivative F defined for all x in that same closed interval:
∫abf(x)dx = F(b) - F(a)
Basically, the second fundamental is a quick and elegant way to solve an integral on a closed interval by simply solving the anti-derivative for its end points. For example, if you're looking for the integral of f(x) = 2x between 1 and 5, you just need an antiderivative; for instance, we can use the simplest such antiderivative F(x) = x
2. With that, you shove both ends of the interval in there to get F(5) - F(1), subtract the start from the end, solve each function (25 - 1), and pop out with 24.
The first fundamental theorem, in case you meant to ask about that, is simply something that tells you that integration and derivation are actually tied together as flip sides of the same thing by way of the anti-derivative, in an analogous sense to multiplication and division or addition and subtraction Basically, if you've got the integral of some blah (f) with respect to some ugh (x), and you take the derivative of that integral with respect to the same ugh, you pop out the original blah. As for what you use it for, if you ever want to clean out an integral to get at the equation inside, such as in solving equations, you can do so by taking the derivative. Or, on the flip side, if you want to get inside of a derivative, take the integral.
Another thing that the first part tells you is that if you've got a continuous function, there is *some* set of functions out there that are anti-derivatives of it. It doesn't even have to be continuous over the entire system of real numbers; as long as there's an interval of continuity, that interval has some set of anti-derivatives. As well, if you ever want to find the area under a line on a chart between two points, that's another use for your anti-derivative. More generally, calculus is used all over the place, and the fundamental theorem of calculus is at its heart.