FToC 1

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,"t"},{5,"b"}},None}],
Show[
Plot[g[x],{x,0.99,t},PlotRange->{{0,6},{0,20}},
Ticks->{{{1,"a"},{t,"t"},{5,"b"}},None}]]
}],
{t,1,5}]