Exercise 25

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. Write an iterated integral whose value represents the volume under the surface $T$ over the rectangle $R$.

2. Evaluate the iterated integral you determined in (a).

3. Find the area of the rectangle, $R$.

4. Determine the exact average temperature, $T_{\operatorname{AVG}(R)}$, over the region $R$.