Question

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.)

Answer 1

The average value of $f$ on $R$ is

$\dfrac{\displaystyle\iint_Rf(x,y)\,\mathrm dA}{\displaystyle\iint_R\mathrm dA}$

i.e. the ratio of the integral of $f$ over $R$ to the measure/area of $R$.

We have

$\displaystyle\iint_R\mathrm dA=\int_0^{\pi/3}\int_0^{\pi/2}\mathrm dx\,\mathrm dy=\frac{\pi^2}6$

and

$\displaystyle\iint_R7\sin x\cos y\,\mathrm dA=7\left(\int_0^{\pi/3}\cos y\,\mathrm dy\right)\left(\int_0^{\pi/2}\sin x\,\mathrm dx\right)=\dfrac{7\sqrt3}2$

So the average value is

$\bar f=\dfrac{\frac{7\sqrt3}2}{\frac{\pi^2}6}=\boxed{\dfrac{21\sqrt3}{\pi^2}}$