Integral of a vector¶
Similarly to the case of the derivative, there are many ways to define the integral of a vector. In this case we will call integral of a vector when the latter depends on a parameter (that in our case will usually be time) that is used to integrate it. In other pages we will discuss how other integrals connect the vector to a particular path where the integral is carried out (line integral of a vector).
Definition¶
Assume that you have a vector \(\vec{f}\) that depends on a parameter t, thus \(\vec{f}=\vec{f}(t)\). The integral of the vector is,
(46)¶\[\begin{split}\vec{F}(t)=\int \vec{f}(t)dt&=\int \left(f_x(t)\vec{i}+f_y(t)\vec{j}+f_z(t)\vec{k}\right)dt\\
&=\left(\int f_x(t)dt\right) \vec{i}+\left(\int f_y(t)dt \right)\vec{j}+\left(\int f_z(t)dt \right)\vec{k}\end{split}\]