We'd do this by breaking the object into infinitesimal pieces of charge, integrating all the contributions of the field from each charge to that point P . However, we were taught to integrate the " components " of the field vector, because simply integrating dE = \ frac { k dq } { r ^ 2 } would not get us the resultant vector that represented the field at that point ( of course, if we did integrate that, we'd be modeling a vector in the positive direction that had the magnitude of the integral, but it would not be the same as the field as vectors involve both magnitude and direction, something normal integration techniques do not give us ).
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