Exercise 28

Consider the solid that lies between the paraboloids $z = g(x,y) = x^2 + y^2$ and $z = f(x,y) = 8 - 3x^2 - 3y^2$.

1. By eliminating the variable $z$, determine the curve of intersection between the two paraboloids, and sketch this curve in the $xy$-plane.

2. Set up, but do not evaluate, an iterated integral expression whose value determines the mass of the solid, integrating first with respect to $z$, then $y$, then $x$. Assume the the solid's density is given by $\delta(x,y,z) = \frac{1}{x^2 + y^2 + z^2 + 1}$.

3. Set up, but do not evaluate, iterated integral expressions whose values determine the mass of the solid using all possible remaining orders of integration. Use $\delta(x,y,z) = \frac{1}{x^2 + y^2 + z^2 + 1}$ as the density of the solid.

4. Set up, but do not evaluate, iterated integral expressions whose values determine the center of mass of the solid. Again, assume the the solid's density is given by $\delta(x,y,z) = \frac{1}{x^2 + y^2 + z^2 + 1}$.

5. Which coordinates of the center of mass can you determine without evaluating any integral expression? Why?