1

f[x_] := E^x Sin[x];

2 Pi Integrate[
        f[x] Sqrt[1 + Evaluate[D[f[x], x]^2]], {x, 0,  Pi}]

2 Pi NIntegrate[
        Evaluate[f[x] Sqrt[1 + D[f[x], x]^2]], {x, 0,  Pi}]

2  Integrate[f[x] ^2, {x, 0,  Pi}]

N[%]

2

f[x_] := E^(-x);

2 Pi Integrate[f[x] Sqrt[1 + Evaluate[D[f[x], x]^2]], {x, 0, a}] // 
      Simplify

2 Pi Integrate[f[x] Sqrt[1 + Evaluate[D[f[x], x]^2]], {x, 0, a}] /. 
      a -> Infinity

Limit[
      (1/2) π (2 (Sqrt[2]) - 
            2 (E^(-a)) (Sqrt[1 + (E^(-2 a))]) + Log[3 + 
                2 (Sqrt[2])] - (((E^a) (Sqrt[1 + (E^(-2 a))]) Log[(E^(-2 a)) (2 + (E^(2 a)) + 2 (Sqrt[1 + (E^(2 a))]))])/(Sqrt[1 + (E^(2 a))]))),  
      a -> Infinity]

N[%]

2  Integrate[f[x] ^2, {x, 0,  a}]

2  Integrate[f[x] ^2, {x, 0,  a}] /. a -> Infinity

3

xS = Pi/2

N[%]

yS = 
      Integrate[
          Sin[x]^2 - (Sin[x]/2)^2, {x, 0,  
            Pi}]/(2 Integrate[
              Sin[x] - (Sin[x]/2), {x, 0,  Pi}])

N[%]