How can i visually or geometrically say that integration is just the inverse of differentiaiton? I have just finished a course on calculus. So I started pondering on the fundamental theorem of calculus and the relation between integration and differentiation . We all know that integration is just the inverse of differentiation. When I look at the graph of a function and look at what the derivative means and the integral between the interval [a,b] means they seem to be logically way too apart from each other. But based on calculations we just say that integration is just the inverse of differentiation. So is this fact only true when we look at calculations or is there even some geometric relation between the two processes  ?
 A: The inverse of differentiation is actually:
$$\int \limits_0^x \frac{\mathrm{d}f(t)}{\mathrm{d}t} \mathrm{d}t = f(x)$$
This means that as long as $\frac{\mathrm{d}f(t)}{\mathrm{d}t}$ is positive, the integral keeps getting bigger, so the function is rising, when $\frac{\mathrm{d}f(t)}{\mathrm{d}t}$ becomes negative the integral starts becoming smaller so the function is descending.
A: I hope that tis two pictures can help.

In the first is represented the area $F(x)$ under the graph of the function $y=f(x)$, starting from $x=0$. When we pass from a point $x$ to $x+dx$ the area increments of a quantity that is approximated by the pink rectangle, i.e. $f(x)dx=F(x+dx)-F(x)=dF(x)$. We can put this intuition in a rigorous statement using the limit notion, as:
$$
\lim_{x \rightarrow 0}\dfrac{dF(x)}{dx}=\lim_{x \rightarrow 0}\dfrac{F(x+dx)-F(x)}{dx} =f(x).
$$
And tis means that  $f(x)$ is the derivative of $F(x)$.
Note that if we start from a point $a \ne 0$ the difference from the two areas is the blue area in the figure, that is a constant value $C$ , and this implies that we have a new primitive function of $f(x)$ : $G(x)=F(x)+C$, in accord with the fact that two functions that differ for a constant have the same derivative $f(x)$.

The second picture express the same result using the relation between a function and his derivative. Starting from $x=a$, where the value of the function is $F(a)$, we can reach the value of the function in $x=t$ adding increments $dF$ to any value $f(x)$ that we reach. This can be expressed as: 
$$
F(t)=F(a)+ \sum _a^t dF(x)
$$
Now note that $dF(x)$ can be approximated as $dF(x)=F'(x)dx$, where $F'(x)$ is the derivative of $F(x)$ at the point $x$, and, using the limit , we have:
$$
F(t)=F(a)+ \lim_{dx \rightarrow 0}\sum _a^t dF(x)=\int_a^t F'(x)dx
$$
Note that, also in this case $F(t)$ is a primitive of a function $f(x)=F'(x)$ , defined less a constant $F(a)$.
