Question

Find the average value fave of the function f on the given interval.

f(t) = esin(t) cos(t), [0, π/2]

Answer 1

The average value of a function over an interval $[a,b]$ is given by

$\displaystyle\frac1{b-a}\int_a^bf(x)\,\mathrm dx$

We have

$\displaystyle\frac1{\frac\pi2-0}\int_0^{\pi/2}e^{\sin t}\cos t\,\mathrm dt=\frac2\pi\int_0^{\pi/2}e^{\sin t}\,\mathrm d(\sin t)$

$=\displaystyle\frac2\pi\int_{\sin0}^{\sin(\pi/2)}e^u\,\mathrm du=\frac2\pi e^u\bigg|_{\sin0}^{\sin(\pi/2)}$

$=\dfrac{2(e^1-e^0)}\pi=\dfrac{2(e-1)}\pi$