Chapter 15.5, Problem 16ES

In some cases it is possible to use Definition 15.5.1 along with symmetry considerations to evaluate a surface integral without reference to a parametrization of the surface. In these exercises, $\sigma$ denotes the unit sphere centered at the origin.

Use the argument in Exercise 15 to prove that if $f\left(x\right)$ is a continuous odd function of $x,$ and if $g\left(y,z\right)$ is a continuous function, then

${\displaystyle \underset{\sigma }{\iint }f\left(x\right)g\left(y,z\right)dS}=0$