f[x_]=(Sqrt[x])

plotf=Plot[f[x],{x,0.1,0.5},PlotStyle->{Thick,Red},AxesLabel->{"x","f(x)"},PlotLegends->{"Начална функция"}]

info={{0.1,f[0.1]},{0.2,f[0.2]},{0.3,f[0.3]},{0.4,f[0.4]},{0.5,f[0.5]}}

plotinfo=ListPlot[info,PlotStyle->{ Blue},AxesLabel->{"x","y"},PlotLegends->{"Пресечни точки"}]

a[{x1_,y1_},{x2_,y2_}]=(y1-y2)/(x1-x2)

b[{x1_,y1_},{x2_,y2_}]=(x1*y2-x2*y1)/(x1-x2)

t[a_,b_,t_]=a*t+b

g[x_]=({{\[Piecewise], ({{t[a[{0.1,f[0.1]},{0.2,f[0.2]}],b[{0.1,f[0.1]},{0.2,f[0.2]}],x], 0.1≤x≤0.2}, {t[a[{0.2,f[0.2]},{0.3,f[0.3]}],b[{0.2,f[0.2]},{0.3,f[0.3]}],x], 0.2≤x≤0.3}, {t[a[{0.3,f[0.3]},{0.4,f[0.4]}],b[{0.3,f[0.3]},{0.4,f[0.4]}],x], 0.3≤x≤0.4}, {t[a[{0.4,f[0.4]},{0.5,f[0.5]}],b[{0.4,f[0.4]},{0.5,f[0.5]}],x], 0.4≤x≤0.5}})}})

plotg=Plot[g[x],{x,0.1,0.5},PlotStyle->Black,AxesLabel->{"x","g(x)"},PlotLegends->{"Начупена крива"}]

Show[plotf,plotinfo,plotg]

N[(∫_0.1^0.5)f[x]\[DifferentialD]x]

N[(∫_0.1^0.5)g[x]\[DifferentialD]x]