Fundamental Theorem of Calculus concept

[Wm_Davies]

I just learned about the fundamental theorem of calculus. I can see that this ties together differentiation and intergration, but I was wondering what kind of problems can be solved by using this theorem? In other words, what can the theorem be applied to?

 

[quasar987]

The definite integral of a function f(x) is a number representing the area under the curve of f. The way it is defined is as a limiting process in which we approximate the area under the curve of f as the sum of the area of little rectangles. Most likely, you have computed the integral of a few functions using this definition. Those were pretty simple functions I'm sure... the likes of polynomials or "step functions". But for most functions it is hard to compute integrals using the definition alone.

And this is where the fundamental theorem of calculus comes into play! It says, well computing the integral of f(x) between a and b is easy: you only have to find a function F(x) such that F'(x) =f(x). Then
$$\int_a^bf(x)dx = F(b)-F(a)$$

Voila!