Question

Find the average value of the function over the given interval. (Round your answer to three decimal places.) f(x) = −sin x, [0, π] Find all values of x in the interval for which the function equals its average value. (Enter your answers as a comma-separated list. Round your answers to three decimal places.)

Answer 1

Answer with Step-by-step explanation:

We are given that

$f(x)=-sin x$

$[0,\infty]$]

Average value of the function is given gy

$f_{avg}=\frac{1}{b-a}\int_{a}^{b}f(x)dx=\frac{1}{\pi-0}\int_{0}^{\pi}-sinx dx$

$f_{avg}=\frac{1}{\pi}[cosx]^{\pi}_{0}$

Using the formula

$\int sin xdx=-cos x$

$f_{avg}=\frac{1}{\pi}(cos\pi-cos0)$

$f_{avg}=\frac{1}{\pi}(-1-1)=-\frac{2}{\pi}$

$f(x)=f_{avg}$

$-sinx=-\frac{2}{\pi}$

$sinx=\frac{2}{\pi}$

$x=sin^{-1}(\frac{2}{\pi})=0.69radian$