Halle el valor medio de la función f(x) = 〖2x〗 ^ 2 - 2x + 2 en el intervalo [0, 8]?

Sabemos que el valor promedio de una función en un

intervalo [a, b] es igual a :

<img src="https://tex.z-dn.net/?f=f%20prom%20%3D%20%5Cfrac%7B1%7D%7Bb-a%7D%5Cint%20_a%5Ebf%5C%3A%5Cleft%28x%5Cright%29dx" />

Aplicandolo en la función dada por el problema :

<img src="https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B8-0%7D%5Cint%20_0%5E8%5Cleft%282x%5E2-2x%2B2%5Cright%29dx" />

Resolvemos la integral :

Aplicamos la regla de la suma :

<img src="https://tex.z-dn.net/?f=%5Cint%20f%5Cleft%28x%5Cright%29%5Cpm%20g%5Cleft%28x%5Cright%29dx%3D%5Cint%20f%5Cleft%28x%5Cright%29dx%5Cpm%20%5Cint%20g%5Cleft%28x%5Cright%29dx" />

<img src="https://tex.z-dn.net/?f=%3D%5Cint%20%5C%3A2x%5E2dx-%5Cint%20%5C%3A2xdx%2B%5Cint%20%5C%3A2dx" />

<img src="https://tex.z-dn.net/?f=%3D%5Cfrac%7B2x%5E3%7D%7B3%7D-x%5E2%2B2x%20%2B%20C" />

Calculamos los límites :

<img src="https://tex.z-dn.net/?f=%5Cint%20_a%5Ebf%5Cleft%28x%5Cright%29dx%3DF%5Cleft%28b%5Cright%29-F%5Cleft%28a%5Cright%29%3D%5Clim%20_%7Bx%5Cto%20%5C%3Ab-%7D%5Cleft%28F%5Cleft%28x%5Cright%29%5Cright%29-%5Clim%20_%7Bx%5Cto%20%5C%3Aa%2B%7D%5Cleft%28F%5Cleft%28x%5Cright%29%5Cright%29" />

<img src="https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5C%3A0%2B%7D%5Cleft%28%5Cfrac%7B2x%5E3%7D%7B3%7D-x%5E2%2B2x%5Cright%2BC%29%3D0" />

<img src="https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5C%3A8-%7D%5Cleft%28%5Cfrac%7B2x%5E3%7D%7B3%7D-x%5E2%2B2x%5Cright%2BC%29%3D%5Cfrac%7B880%7D%7B3%7D" />

<img src="https://tex.z-dn.net/?f=%3D%5Cfrac%7B880%7D%7B3%7D-0%3D%5Cfrac%7B880%7D%7B3%7D" />

<img src="https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B8-0%7D%5Ccdot%20%5Cfrac%7B880%7D%7B3%7D" />

[img = 10]

El valor promedio es igual a [img = 11].