Cumulative Area Function

The graph on the left shows f(x)as well as the area A(x) under f from a to x.  The graph on the right is of the cumulative area under f, that is A(x).

f[x_]:=2ArcTan[x-3]+3;
F[t_]=∫f[t] \[DifferentialD]t;
g[x_]=Simplify[F[x]-F[1]];

Manipulate[
GraphicsRow[{
Show[
Plot[f[x],{x,0,6},Axes->True,PlotRange->{{0,6},{0,6}},Ticks->False],
RegionPlot[y<f[x],{x,.99,t},{y,0,6},Axes->True,PlotRange->{{0,6},{0,6}}],
Ticks->{{{1,"a"},{t,"x"},{5,"b"}},None},PlotLabel->"The Function f(x)"],
Show[
Plot[g[x],{x,0.99,t},PlotRange->{{0,6},{0,20}},
Ticks->{{{1,"a"},{t,"x"},{5,"b"}},None},PlotLabel->"Cumulative Area A(x)"]]
}],
{{t,2,"x"},1,5}]