Finding the Average Value of a Function

Find the average value of \(f(x)=3x-8\) on the closed interval \([0,2] .\)

Solution The average value of \(f(x)=3x-8\) on the closed interval \([0,2]\) is given by \[ \begin{eqnarray*} \bar{y} &=&\dfrac{1}{b-a}\int_{a}^{b}f(x)\,dx=\dfrac{1}{2-0}{\int_{0}^{2}}(3x-8) dx\\ &=& \dfrac{1}{2}\left[\dfrac{3x^{2}}{2}-8x \right] _{0}^{2} =\dfrac{1}{2}(6-16) =-5 \end{eqnarray*} \]

The average value of \(f\) on \([0,2] \) is \(\bar{y}=-5.\)