Show that the average distance is …, from the North Pole (…)…

Problems, on the attraction of a sphere, use Figure 14.18 and the law of cosines $q^2=D^2-2\rho D\cos \varphi +{\rho }^2=u$.

Show that the average distance is $\bar{q}=4R/3$, from the North Pole ($D=R$) to points on the Earth's surface ($\rho =R$). To compute: $\bar{q}=\iint q{R}^2\sin \varphi d\varphi d\theta /(\text{area }4\pi R^2)$. Use the same substitution $u$.

Its source prints no answer for this problem, so there's nothing to check against.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §14.4 Cylindrical and Spherical Coordinates, #29