Average Value of a Function

Averages weigh by area

Two worked averages

The mean value guarantee

Weighted averages with density

What is the average value of $x y$ over the unit square $[0,1] \times [0,1]$?

Why is the average of $x^2$ over the unit disk smaller than over the interval $[-1, 1]$?

A continuous function on a connected closed region has average value $\bar f$. What is guaranteed?

The average of $x^2$ over the unit disk is $\bar f = c/4$ for some constant... give the average directly as a fraction like 1/3.

Over a region of total mass $6$, the integral $\iint f\, \delta\, dA$ returns $15$. Give the weighted average of $f$, as a decimal like 2.5.