Exercise 25

Consider the plane generated by $z = f(x,y) = 24 - 2x - 3y$ over the region $D = [0,2]\times[0,3]$.

1. Sketch a picture of the overall solid generated by the plane over the given domain.

2. Determine a parameterization $\vr(s,t)$ for the plane over the domain $D$.

3. Use Equation to determine the surface area generated by $f$ over the domain $D$.

4. Observe that the vector $\vu = \langle 2, 0, -4 \rangle$ points from $(0,0,24)$ to $(2,0,20)$ along one side of the surface generated by the plane $f$ over $D$. Find the vector $\vv$ such that $\vu$ and $\vv$ together span the parallelogram that represents the surface defined by $f$ over $D$, and hence compute $| \vu \times \vv |$. What do you observe about the value you find?