How can I find the average y value of a function on a given domain?

Question

Lets say that $f(x) = (10 - x)\ln x$. Over the domain: $1 ≤ x ≤ 10$. How can I find the average value of $y$ over this domain and what is that value?

Answer 1

To find the average $\bar f$ of a function $f \colon [a,b] \to \mathbb R$ calculate $$ \bar f = \frac 1{b-a} \int_a^b f(x)\, dx $$ In your case, we have \begin{align*} \bar f &= \frac 1{10-1} \int_1^{10} (10 - x)\log x \, dx \\ &= \frac 1{10 - 1} \left[10x(\log x - 1) - \frac 14x^2(2\log x - 1)\right]_1^{10}\\ &= \frac 19 \left(100(\log 10 - 1) - 25(2\log 10 - 1) - 10(-1) + \frac 14(-1)\right)\\ &= \frac 19 \left(50\log 10 - 65 - \frac 14\right) \end{align*}

Answer 2

The mean of an integrable function $\,f(x)\,$ over an interval $\,[a,b]\,$ is given by

$$\overline f:=\frac{1}{b-a}\int_a^bf(x)\,dx$$