Exercise 15

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. Determine the area of the rectangle $R$.

3. Estimate the average temperature, $T_{\operatorname{AVG}(R)}$, over the region $R$.

4. Do you think your estimate in (c) is an over- or under-estimate of the true temperature? Why?