Question

Find the average value of f over the given rectangle. f(x, y) = 3x2y, r has vertices (−5, 0), (−5, 3), (5, 3), (5, 0). g

Answer 1

$\dfrac{\displaystyle\iint_Rf(x,y)\,\mathrm dx\,\mathrm dy}{\displaystyle\iint_R\mathrm dx\,\mathrm dy}$
$\dfrac{\displaystyle\int_{y=0}^{y=3}\int_{x=-5}^{x=5}3x^2y\,\mathrm dx\,\mathrm dy}{\displaystyle\int_{y=0}^{y=3}\int_{x=-5}^{x=5}\mathrm dx\,\mathrm dy}$
$\displaystyle\frac1{10}\left(\int_{y=0}^{y=3}y\,\mathrm dy\right)\left(\int_{x=-5}^{x=5}x^2\,\mathrm dx\right)$
$\dfrac{75}2$