A solid \(S\) is bounded below by the square \(z=0\text{,}\)\(-1 \leq x \leq 1\text{,}\)\(-1 \leq y \leq 1\) and above by the surface \(z = 2-x^2-y^2\text{.}\) A picture of the solid is shown in Figure 12.6.9.
First, set up both 1) an iterated double integral to find the volume of the solid \(S\) as a double integral of a solid under a surface and 2)an iterated triple integral that gives the volume of the solid \(S\text{.}\) You do not need to evaluate either integral. Write a couple of sentences to compare these two approaches.
Set up (but do not evaluate) iterated integral expressions that will tell us the center of mass of \(S\text{,}\) if the density at point \((x,y,z)\) is \(\delta(x,y,z)=x^2+1\text{.}\)
Use technology appropriately to evaluate the iterated integrals you determined in (a), (b), and (c); does the location you determined for the center of mass make sense?
