f[x_]=Log[x]

plotf=Plot[f[x],{x,1,2},PlotStyle->{Thick,Red},AxesLabel->{"x","f(x)"},PlotLegends->{"Начална функция"}]

info={{1,f[1]},{1.4,f[1.4]},{1.8,f[1.8]},{2,f[2]}}

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[{1,f[1]},{1.4,f[1.4]}],b[{1,f[1]},{1.4,f[1.4]}],x], 1≤x≤1.4}, {t[a[{1.4,f[1.4]},{1.8,f[1.8]}],b[{1.4,f[1.4]},{1.8,f[1.8]}],x], 1.4≤x≤1.8}, {t[a[{1.8,f[1.8]},{2,f[2]}],b[{1.8,f[1.8]},{2,f[2]}],x], 1.8≤x≤2}})}})

plotg=Plot[g[x],{x,1,2},PlotStyle->Black,AxesLabel->{"x","g(x)"},PlotLegends->{"Начупена крива"}]

Show[plotf,plotinfo,plotg]

N[(∫_1^2)f[x]\[DifferentialD]x]

N[(∫_1^2)g[x]\[DifferentialD]x]