Exercise 16

Let $f$ be a function of independent variables $x$ and $y$ that is increasing in both the positive $x$ and $y$ directions on a rectangular domain $R$. For each of the following situations, determine if the double Riemann sum of $f$ over $R$ is an overestimate or underestimate of the double integral $\iint_R f(x,y) \, dA$, or if it impossible to determine definitively. Provide justification for your responses.

1. The double Riemann sum of $f$ over $R$ where $f$ is evaluated at the lower left point of each subrectangle.

2. The double Riemann sum of $f$ over $R$ where $f$ is evaluated at the upper right point of each subrectangle.

3. The double Riemann sum of $f$ over $R$ where $f$ is evaluated at the midpoint of each subrectangle.

4. The double Riemann sum of $f$ over $R$ where $f$ is evaluated at the lower right point of each subrectangle.