# Integration as Inverse of Differentation

In the previous chapter we observed that integration of a function $v(t)$ followed by differentiation, gives back the function $v(t)$, which we refer to as The Fundamental Theorem of Calculus:

• $\frac{d}{dt}\int_0^tv(s) ds = v(t)\quad\mbox{for }t>0$.   (1)

Alternatively, The Fundamental Theorem of Calculus can be expressed as

• $\int_0^t\dot u(s) ds = u(t)\quad\mbox{for }t>0$,    (2)

stating: Integration of the derivative $\dot u(t)$ of the function $u(t)$, gives back the function $u(t)$. This follows from the fact that the derivative with respect to $t$ of both sides of (2) equals $\dot u(t)$, combined with the fact that two functions with the same derivative taking the same value for $t=0$, must coincide. Two cars traveling with the same velocitystarting at the same time from the same location will arrive at the same time to the destination. Right?

# The Whole = Sum of Parts

We shall see that (2) can be viewed to express the following identity:

• The sum  ($\int_0^t \mbox{or }\sum_{m=0}^n$) of the parts ($du=\dot uds$) =  the whole ($u(t)$).

Integration means summing little pieces to make up the whole. In Leibniz notation this is expressed as

• $\int_0^t \frac{du}{ds}ds=\int_0^t du =u(t)-u(0)$.

Elementary and profound.

Below we shall study the dependence of the integral $\int_0^t v(s) ds$ of a given function $v(s)$ on the time step $ds$, and see that it is a uniquely determined number for vanishing time step, which is approximated using a finite time step, with accuracy depending on the variation of the function $f(t)$ with $t$. We will thus give a mathematical analysis of the meaning of the Fundamental Theoreom of Calculus, which we will refer to as a mathematical proof of the Fundamental Theorem.

This experience will illustrate the role and meaning of a mathematical proof as a process of dissecting the structure and meaning of a certain mathematical statement.

Analog mechanical integrator computing the integral $\int y(x)dx$ of a funtion $y(x)$. Can you explain how it works?

• Suppose $u(T)=u(0)=0$. What then about $\int_0^Tu^\prime (t)\, dt$?