Chapter 14.8, Problem 32ES

The accompanying figure on the next page shows the solid that is bounded above by the surface $z=1\text{ /}\left(x^2\text{+}{y}^2+1\right),$ below by the xy-plane, and laterally by the surface $x^2+y^2=a^2.$

(a) By symmetry, the centroid of the solid lies on the z-axis. Make a conjecture about the behavior of the z-coordinate of the centroid as $a\to 0^{+}\text{and}\text{as }a\to +\infty .$

(b) Find the z-coordinate of the centroid, and check your conjecture by calculating the appropriate limits.

(c) Use a graphing utility to plot the z-coordinate of the centroid versus a, and use the graph to estimate the value of a for which the centroid is $\left(0,0,0.25\right).$