Let S be the sphere of radius R. a. Argue by symmetry that

∫ ∫ x^2dS = ∫ ∫ y^2 dS= ∫ ∫ z^2dS
S S S

b. ) Use this fact and some clever thinking to evaluate, with very little computation, the integral

∫∫x^2 dS.
S

Step-by-step explanation:

a. Given that the sphere is symmetrical in nature. If we take a look at the yz-plane, over the hemisphere,  the integral on the surface of x on both sides of the plane will annul each other and the outcome will be zero. However, the remaining two integrals will also be zero as a result of symmetry on the xy and xz planes.

Thus,

b. Recall that :

By applying the integral of the surface area.

      (surface area of a sphere)

From above;

∴