Why are the fundamental theorems of calculus usually associated to the Riemann Integral?

Solution 1:

Here is my opinion. When I studied the FTC, I indeed associate it with Riemann integrals because the following theorem presented in Rudin's principle of mathematical analysis (Thereom 6.21):

If $F'=f$ for some Riemann integrable function $f$ over $[a,b]$, then $$ F(b)-F(a)=\int_a^b f(x)dx $$ Its proof is quite natural.

We consider a partition $P:\{x_0=a\leq x_1\leq x_2\cdots\leq x_n=b\}$, and apply the mean value theorem $$ F(b)-F(a)=\sum_{i=1}^nF(x_i)-F(x_{i-1})=\sum_{i=1}^n f(t_{i}) (x_i-x_{i-1}). $$ for $t_i\in [x_{i-1},x_i]$. Given any $\epsilon$, the partition $P$ can be choosen so that $$ |\sum_{i=1}^n f(t_{i}) (x_i-x_{i-1})-\int_a^b f(t)ds|\leq \epsilon. $$ Thus, we complete the proof. I think this proof links the Riemann integral and FTC in the best sense.