Chapter 14.2, Problem 61ES

The average value or mean value of a continuous function $f(x,y)$ over a region R in the $xy\text{-plane}$ is defined as $f_{ave}=\frac{1}{A(R)}{\displaystyle \underset{R}{\iint }f(x,y)dA}$ where $A(R)$ is the area of region R (compare to the definition preceding Exercise 35 in Section 14.1). Use this definition in these exercises.

Find the average value of $1/(1+x^2)$ over the triangular region with vertices $(0,0),(1,1),\text{and}\text{(0,1)}\text{.}$