Can't understand Second Fundamental Theorem of Calculus sorry if this has been asked before, but I can't seem to find my question in particular.
Anyway, in the Second FTM it says $$F(x)=\int_a^xf(x)dx$$
If I understand correctly is just the area under the curve. no problem there.
Then it says that$$\int_a^bf(x)dx=F(b)-F(a)$$
if I'm thinking this correctly, it would make sense because in  $F(a)=\int_a^af(x)dx$
 the area is $0$ so I'm just left with $F(b)$ which is the integral from a to b, this is where I think I have to be wrong, because in every example I see, they take the value of $a$ and plug it in $F(x)$.
For example in $$\int_2^5x^2dx$$ with $F(x)=x^3/3$
 they take $F(5)$ and $F(2)$, in particular $F(2)=2^3/3$, but shouldn't $F(2)$ always be $0$ because it's basically just $\int_2^2x^2dx$?
p.s Sorry in advance for any mistake I made in formating or anything else.


Here are some aspects which might be helpful
  
  
*
  
*Be careful about not mixing up in the first part of FTC the variable of integration with the variable $x$. 
  \begin{align*}
F(x)=\int_a^xf(t)\,dt\tag{1}
\end{align*}
  The variable $t$ in (1) is a bound variable similar as the index $k$ in a sum $\sum_{k=0}^nf(k)$, while $x$ is a free variable.
  
*The function $F$ is not the area under the curve. The area is a number while $F$ is not a number but a function in $x$.
But, when we evaluate the function $F$ at a specific value $x=b$ we obtain the number
  \begin{align*}
F(b)=\int_{a}^bf(t)\,dt+F(a)
\end{align*}
  resp.
  \begin{align*}
\int_{a}^bf(t)\,dt=F(b)-F(a)\tag{2}
\end{align*}
  and $F(b)-F(a)$ in (2) can be interpreted as (signed) area under the curve in $[a,b]$.

