Exercise 23

Consider the solid region $S$ bounded above by the paraboloid $z = 16 - x^2 - y^2$ and below by the paraboloid $z = 3x^2 + 3y^2$.

1. Describe parametrically the curve in $\R^3$ in which these two surfaces intersect.

2. In terms of $x$ and $y$, write an equation to describe the projection of the curve onto the $xy$-plane.

3. What coordinate system do you think is most natural for an iterated integral that gives the volume of the solid?

4. Set up, but do not evaluate, an iterated integral expression whose value is average $z$-value of points in the solid region $S$.

5. Use technology to plot the two surfaces and evaluate the integral in part d. Write at least one sentence to discuss how your computations align with your intuition about where the average $z$-value of the solid should fall.