Answer
$${2 \over 5\pi}$$
Work Step by Step
let $u = cos x$
$$h_{ave} = {1 \over \pi} \int ^{\pi} _0 cos^4 x sin x dx = {1 \over \pi} \int ^{-1} _{1} u^4 (-du)$$
$${1 \over \pi} 2 \int ^1_0 u^4 du = {2 \over \pi} [{1\over 5} u^5] ^1 _0 = {2 \over 5\pi}$$
