Compute the average value of the following function over the region R. f(x,y) = 7 sin x cos y R= { (x,y): 0 leq x leq pi/2, 0 leq y leq pi/3} bar f = (Simplify your answer. Type an exact answer, using radicals as needed. Type your answer in factored form. Use integers or fractions for any numbers in the expression.)

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]