Double Riemann Sums and Double Integrals over Rectangles

The temperature at any point on a metal plate in the $xy$ plane is given by $T(x,y) = 100-4x^2 - y^2$, where $x$ and $y$ are measured in inches and $T$ in degrees Celsius. Consider the portion of the plate that lies on the rectangular region $R = [1,5] \times [3,6]$.

1. Estimate the value of $\iint_R T(x,y) \, dA$ by using a double Riemann sum with two subintervals in each direction and choosing $(x_i^*, y_j^*)$ to be the point that lies in the upper right corner of each subrectangle.

2. Determi…

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…

Consider the box with a sloped top that is given by the following description: the base is the rectangle $R = [0,4] \times [0,3]$, while the top is given by the plane $z = p(x,y) = 20 - 2x - 3y$.

1. Estimate the value of $\iint_R p(x,y) \, dA$ by using a double Riemann sum with four subintervals in the $x$ direction and three subintervals in the $y$ direction, and choosing $(x_i^*, y_j^*)$ to be the point that is the midpoint of each subrectangle.

2. What important quantity does your double R…