#22, 36, 39, pp. 567

f[x_]:=x+Cos[x]
fp[x_]=f'[x]
Plot[f[x],{x,0,Pi/2}]
Plot[Sqrt[1+fp[x]^2],{x,0,Pi/2}]
Simplify[Sqrt[1+fp[x]^2]]
MidpointApproximation=N[Sqrt[1+fp[Pi/4]^2]*Pi/2]
1.05*Pi/2
NIntegrate[Sqrt[1+fp[x]^2],{x,0,Pi/2}]

f[x_]:=150-1/40*(x-50)^2
Plot[f[x],{x,0,80}]
fp[x_]=f'[x]
Simplify[Sqrt[1+fp[x]^2]]
NIntegrate[Sqrt[1+fp[x]^2],{x,0,80}]

f[x_]:=Sin[Pi*x/7]
Plot[f[x],{x,0,28}]
fp[x_]=f'[x]
Simplify[Sqrt[1+fp[x]^2]]
NIntegrate[Sqrt[1+fp[x]^2],{x,0,28}]