Chapter 15.1, Problem 3PT

Let $R=\left\{\left(x,y\right)|3\le x\le 5,2\le y\le 4\right\}\text{and}f\left(x,y\right)=y^2-x^2$.For every partition of R, the double Riemann sum for ${\displaystyle \underset{R}{\iint }\left(y^2-x^2\right)dA}$ will be largest if, for each subrectangle, we choose $\left(x_i^{*},y_j^{*}\right)$ so that:

1. a) $x_i^{*}$ is the left (lower) endpoint and $y_j^{*}$ is the left (lower) endpoint.
2. b) $x_i^{*}$ is the left (lower) endpoint and $y_j^{*}$ is the right (upper) endpoint.
3. c) $x_i^{*}$ is the right (upper) endpoint and $y_j^{*}$ is the left (lower) endpoint.
4. d) $x_i^{*}$ is the right (upper) endpoint and $y_j^{*}$ is the right (upper) endpoint.