Find the average value of f over region d. f(x, y) = 7xy, d is the triangle with vertices (0, 0), (1, 0), and (1, 8).

is a right triangle with base length 1 and height 8, so the area of is .

The average value of over is given by the ratio



The denominator is just the area of , which we already know. The average value is then simplified to



In the -plane, we can describe the region as all points that lie between the lines and (the lines which coincide with the triangle's base and hypotenuse, respectively), taking . So, the integral is given by, and evaluates to,