p. 568

41.

The average value of a function f(x) on an interval [a, b] is given by \(TraditionalForm
      \`avg(f)\  = \ 
        \(1\/\(b - a\)\) 
          \(∫\_a\%b\( f(x)\) dx\)\). So the average value of f(t)  =  2  (e^(-t)) cos(t) on [0, 2  π] is \(TraditionalForm
      \`\(1\/\(2  π\)\) 
        \(∫\ 2 \( e\^\(-t\)\) \(cos(t)\) dt\)\). Using integration by parts (similar to problem 21), we get avg(f)  =  (1/(2  π))(e^(-t))(sin(t)  -  cos(t))\(|\_0\%\(2  π\)\)= (1/(2  π)) (1  -  (e^(-2 π))).

(1/(2  π)) 
      ∫ 2  (E^(-t)) Cos[t] \[DifferentialD]t

(1/(2  π)) 
      (∫_0^(2  π)) 
        2  (E^(-t)) Cos[t] \[DifferentialD]t

N[((1 - (E^(-2 π)))/(2 π))]

\(Plot[{2 \( E\^\(-t\)\) Cos[t], \ 
        \(1\/\(2  π\)\) \((1\  - \ E\^\(\(-2\) π\))\)}, {t, 0, 
        2  π}, 
	PlotStyle -> {RGBColor[0, 0, 1], \ RGBColor[1, 0, 0]}, 
      \ PlotRange -> {\(-1\), 2}]\)