Question

Find the Mean value of the function f left parenthesis x right parenthesis space equals space x cubed space minus space 3 x squared over the interval [0, 5].

Answer 1

Let us find the Mean value of the function $f(x)=x^3 - 3 x^2$ over the interval $[0, 5].$

It follows that the Mean value of the function is

$\overline{f}=\frac{1}{5-0}\int\limits_0^5f(x)dx =\frac{1}{5}\int\limits_0^5f(x^3-3x^2)dx =\frac{1}{5}(\frac{x^4}4-x^3)|_0^5\\ =\frac{1}5(\frac{625}4-125)=\frac{1}5\cdot\frac{125}4=\frac{25}{4}=6.25$