The average value of [tex]f[/tex] on [tex]R[/tex] is
[tex]\dfrac{\displaystyle\iint_Rf(x,y)\,\mathrm dA}{\displaystyle\iint_R\mathrm dA}[/tex]
i.e. the ratio of the integral of [tex]f[/tex] over [tex]R[/tex] to the measure/area of [tex]R[/tex].
We have
[tex]\displaystyle\iint_R\mathrm dA=\int_0^{\pi/3}\int_0^{\pi/2}\mathrm dx\,\mathrm dy=\frac{\pi^2}6[/tex]
and
[tex]\displaystyle\iint_R7\sin x\cos y\,\mathrm dA=7\left(\int_0^{\pi/3}\cos y\,\mathrm dy\right)\left(\int_0^{\pi/2}\sin x\,\mathrm dx\right)=\dfrac{7\sqrt3}2[/tex]
So the average value is
[tex]\bar f=\dfrac{\frac{7\sqrt3}2}{\frac{\pi^2}6}=\boxed{\dfrac{21\sqrt3}{\pi^2}}[/tex]
