Chapter 5.6, Problem 11MP

As needed, use a computer to plot graphs and to check values of integrals.

Show that the z coordinate of the centroid of the volume inside the elliptic cone $\frac{z^2}{h^2}=\frac{x^2}{a^2}+\frac{y^2}{b^2},0<z<h\text{, is }\bar{z}=\frac{3}{4}h.$

(Note that the result is independent of a and b.) Hint: To evaluate the triple integrals, let $z=hz\text{'},x=ax\text{'},y=by\text{'}$, and then change to cylindrical coordinates in the primed system (see Example 4, Section 4). Compare Example 2, Section 4.