user friendly proof of fundamental theorem of calculus Silly question. Can someone show me a nice easy to follow proof on the fundamental theorem of calculus. More specifically,
$\displaystyle\int_{a}^{b}f(x)dx = F(b) - F(a)$
I know that by just googling fundamental theorem of calculus, one can get all sorts of answers, but for some odd reason I have a hard time following the arguments. 

i will assume that $f$ is continuous, $F$ is differentiable in $(a,b)$ and $$F'(x)=f(x).  $$  we will need the mean value theorem in the form $$\text{ for any  } a \le c, d < \le b,\quad F(d) - F(c) = (d-c)F'(t) = (d-c)f(t) \text{ for some } c < t < d.$$
let $a \le x_0 \le x_1\le \cdots\le x_n = b$ any partition of $[a, b].$ then $$\begin{align}F(b) - F(a) &= \left(F(x_n) - F(x_{n-1}\right) + \cdots  +\left(F(x_j) - F(x_{j-1}\right)+\cdots+\left(F(x_1) - F(x_{0}\right)\\
&=(x_n-x_{n-1})f(t_n)+\cdots+(x_j-x_{j-1})f(t_j)+\cdots+(x_1-x_0)f(t_1)\\
&=\int_a^bf(x)\, dx.\end{align} $$
