Answer
$${V_{avg}} = \frac{7}{{2\pi }}{\text{ liters}}$$
Work Step by Step
$$\eqalign{
& {\text{The volume is given by }} \cr
& V = 1.75\sin \frac{{\pi t}}{2},{\text{ }}t{\text{ in seconds}} \cr
& {\text{Let }}V = 0{\text{ to find the interval of the positive cycle}} \cr
& 1.75\sin \frac{{\pi t}}{2} = 0 \cr
& {\text{Solving the equation for the interval }}\left[ {0,2} \right],{\text{ we obtain}} \cr
& t = 0,{\text{ }}t = 2 \cr
& \cr
& {\text{The average volume of air is given by:}} \cr
& {V_{avg}} = \frac{1}{{2 - 0}}\int_0^2 {\left( {1.75\sin \frac{{\pi t}}{2}} \right)} dt \cr
& {V_{avg}} = \frac{{1.75}}{2}\int_0^2 {\sin \frac{{\pi t}}{2}} dt \cr
& {V_{avg}} = \frac{7}{8}\int_0^2 {\sin \frac{{\pi t}}{2}} dt \cr
& {\text{Integrating}} \cr
& {V_{avg}} = \frac{7}{8}\left( {\frac{2}{\pi }} \right)\left[ { - \cos \left( {\frac{{\pi t}}{2}} \right)} \right]_0^2 \cr
& {V_{avg}} = - \frac{7}{{4\pi }}\left[ {\cos \left( {\frac{{2\pi }}{2}} \right) - \cos \left( {\frac{0}{2}} \right)} \right] \cr
& {V_{avg}} = - \frac{7}{{4\pi }}\left( { - 1 - 1} \right) \cr
& {V_{avg}} = \frac{7}{{2\pi }}{\text{ liters}} \cr} $$
