f[x_] = 2x - x^2;
Manipulate[
Show[
ParametricPlot[{{c1 + .01Cos[t], .01 Sin[t]}, {c2 + .01 Cos[t], .01Sin[t]}, {c3 + .01 Cos[t], .01Sin[t]}, {c4 + .01 Cos[t], .01Sin[t]}}, {t, 0, 2 Pi}, PlotRange -> {{-.25, 2.25}, {-.25, 1.25}}, PlotStyle -> {{Thickness[.02], Red},{Thickness[.02], Red},{Thickness[.02], Red},{Thickness[.02], Red}}],
ParametricPlot[{{c1, t f[c1]}, {c2, t f[c2]}, {c3, t f[c3]}, {c4, t f[c4]}}, {t, 0, 1}, PlotRange -> {{-.25, 2.25}, {-.25, 1.25}}, PlotStyle -> {Red, Red, Red, Red}],
ParametricPlot[{{0, -.1 t}, {1/2, -.1 t},{1, -.1 t},{1.75, -.1 t},{2, -.1 t}}, {t, 0, 1}, PlotStyle -> {{Thick, Darker[Green]},{Thick, Darker[Green]},{Thick, Darker[Green]},{Thick, Darker[Green]}, {Thick, Darker[Green]}}],
Plot[
{2x - x^2,
Piecewise[{{0, x< 0}, {f[c1], x < 1/2}, {f[c2], x < 1},{ f[c3], x < 7/4}, {f[c4], x < 2}, {0, x > 2}}] }, {x,-.25, 2.25}, PlotRange -> {-.15, 1.25}, AspectRatio -> Automatic, PlotStyle -> {{Thick, Black},Blue} , Filling ->{2 -> Axis}, FillingStyle -> Directive[Blue, Opacity[.25]]
]
], {c1, 0, .5}, {c2, .5, 1}, {c3, 1, 1.75}, {c4, 1.75, 2}]