I'd like to try and provide a somewhat intuitive explanation of this theorem, since, if you're anything like me, when you first learn it, there seems to be nothing at all intuitive about it.
So, say we have some integrable function f defined on the real numbers. Then, if we assume f is positive, recall that the definite integral of f from a to b, written
| f(t) dt
can be thought of as the area under f
between a and b. OK, then consider the function F defined as follows:
F(x):= | f(t) dt
where x > a. Then F(x) can be thought of as giving us the "area so far" under f from a to x. We can see this graphically (where the "curve" is the graph of the function f):
| ..--' \ __.--- f
| /* '--'
|__.' * *
| * F(x) *
| * *
Now the Fundamental Theorem of Calculus tells us that the derivative
of F at x (let's call it F'(x) ), is equal to f(x). This does not seem obvious at first, but let's see if we can get an idea of why this is so. Remember that the derivative of F at x is defined to be the limit
as h goes to zero
of ((F(x+h)-F(x))/h); that is:
Now, consider the quantity ((F(x+h)-F(x))/h) for some small value of h. Now, F(x+h)-F(x) gives us the area under f between x and x+h:
* F(x+h) - *
* F(x) *
<---- h ---->
But since h is small, we can approximate
this area by the area of a rectangle
with a height of f(x) and width h. So, we have the approximate equality
, h*f(x) ~= F(x+h)-F(x). Dividing both sides by h, we have f(x) ~= (F(x+h)-F(x))/h. Now, if you make h smaller and smaller, then it seems like it would make sense that this approximation would get better and better. So, we haven't proven it, but I hope it's at least believable that when we take the limit as h goes to zero, the approximate equality above becomes actual equality, and we know (by definition) that the right hand side of the equality becomes F'(x), so f(x)=F'(x).
I feel like this makes it a little bit easier to understand what exactly is going on with the Fundamental Theorem (for proof, go to proof of the Fundamental Theorem of Calculus), but I'm open to any suggestions on how to make it more clear. /msg me!