Section 1.4 Vector Integration
Integration is the reverse process of differentiation. Thus, if \(\frac{d}{dt}\vec{F}(t) =\vec{v}(t)\) then,
\begin{equation*}
\int\vec{v}(t)dt=\int \frac{d}{dt}\vec{F}(t) dt=\vec{F}(t)+c
\end{equation*}
where the constant \(c\) is known as the constant of integration. The value of \(c\)is determined by the initial or the geometrical conditions. The above integral is called an indefinite integral of \(\vec{v}(t)\) with respect to \(t\text{.}\) The integral between the limits \(t = t_{1}\) to \(t_{2}\text{,}\) can be written as
\begin{equation*}
\int^{t_{2}}_{t_{1}}\left(\vec{v}(t) dt\right)= \left[\vec{F}(t)+c\right]^{t_{2}}_{t_{1}}= \vec{F}(t_{2})-\vec{F}(t_{1})
\end{equation*}
and is called the definite integral of \(\vec{v}(t)\text{.}\) It is also defined as the limit of the sum.
