Chapter 14.1, Problem 38ES

The average value or mean value of a continuous function $f\left(x,y\right)$ $R=\left[a,b\right]\times \left[c,d\right]$ is defined as $f_{\text{ave}}=\frac{1}{A\left(R\right)}{\displaystyle \underset{R}{\iint }f\left(x,y\right)dA}$

Where $A\left(R\right)=\left(b-a\right)\left(d-c\right)$ is the area of the rectangle $R$ (compare to Definition 5.8.1). Use this definition in these exercises.

Find the average value of $f\left(x,y\right)=x{\left(x^2+y\right)}^{1/2}$ over the rectangle $\left[0,1\right]\times \left[0,3\right].$