Answer
$${f_{{\text{ave}}}} = \frac{\pi }{3}$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = \frac{1}{{\sqrt {1 - {x^2}} }};\,\,\,\left[ { - \frac{1}{2},0} \right] \cr
& {\text{Find the average value using the definition }}{f_{{\text{ave}}}} = \frac{1}{{b - a}}\int_a^b {f\left( x \right)dx} \cr
& {f_{{\text{ave}}}} = \frac{1}{{0 - \left( { - 1/2} \right)}}\int_{ - 1/2}^0 {\frac{1}{{\sqrt {1 - {x^2}} }}dx} \cr
& {f_{{\text{ave}}}} = 2\int_{ - 1/2}^0 {\frac{1}{{\sqrt {1 - {x^2}} }}dx} \cr
& {\text{Integrating }} \cr
& {f_{{\text{ave}}}} = 2\left( {{{\sin }^{ - 1}}x} \right)_{ - 1/2}^0 \cr
& {f_{{\text{ave}}}} = 2\left( {{{\sin }^{ - 1}}0 - {{\sin }^{ - 1}}\left( { - 1/2} \right)} \right) \cr
& {\text{Simplifying}} \cr
& {f_{{\text{ave}}}} = 2\left( {0 - \left( { - \pi /6} \right)} \right) \cr
& {f_{{\text{ave}}}} = 2\left( {\frac{\pi }{6}} \right) \cr
& {f_{{\text{ave}}}} = \frac{\pi }{3} \cr} $$
