Let $R$ be the rectangle with vertices $(0,0)$, $(2,0)$, $(2,2)$, and $(0,2)$ and let $f(x,y) = \sqrt{2xy}$.

(a) Find reasonable upper and lower bounds for $\int_{R}f\,dA$ without subdividing $R$.
upper bound =
lower bound =

(b) Estimate $\int_{R}f\,dA$ three ways: by partitioning $R$ into four subrectangles and evaluating $f$ at its maximum and minimum values on each subrectangle, and then by considering the average of these (over and under) estimates.
overestimate: $\int_{R}f\,dA \approx$
underestimate: $\int_{R}f\,dA \approx$
average: $\int_{R}f\,dA \approx$