Chapter 15.5, Problem 17ES

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.

(a) Explain why

${\displaystyle \underset{\sigma }{\iint }{x}^{2}dS={\displaystyle \underset{\sigma }{\iint }{y}^{2}dS={\displaystyle \underset{\sigma }{\iint }{z}^{2}dS}}}$

(b) Conclude from part (a) that

${\displaystyle \underset{\sigma }{\iint }{x}^{2}dS=\frac{1}{3}\left[{\displaystyle \underset{\sigma }{\iint }{x}^{2}dS+{\displaystyle \underset{\sigma }{\iint }{y}^{2}dS}+{\displaystyle \underset{\sigma }{\iint }{y}^{2}dS}}\right]}$

(c) Use part (b) to evaluate

${\displaystyle \underset{\sigma }{\iint }{x}^{2}dS}$

without performing an integration.