Exercise 18

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 Riemann sum in (a) estimate?

3. Suppose it can be determined that $\iint_R p(x,y) \, dA = 138$. What is the exact average value of $p$ over $R$?

4. If you wanted to build a rectangular box (with the same base) that has the same volume as the box with the sloped top described here, how tall would the rectangular box have to be?