<s>[INST] Determine the average value of the following function Please bear with me because I have only little experience in using codes to construct the symbols for the equations.
The question is:
Determine the average value of $f(x,y) = x^2 y^2$, in the region $$R: a\le x\le b,  c\le y\le d,$$
where $a+b=5, ab=13, c+d=4, cd=7.$
The formula for average value is
$$\dfrac{\displaystyle\iint f(x,y)dA}{(b-a)(d-c)}.$$
Please use only elementary calculus, and no complex analysis. Much appreciated!
 [/INST] I guess this example could be a weird way to show possible transformation formalities during (double-)integration. Let's just do it formally:
$$\dfrac{\displaystyle\iint f(x,y)dA}{(b-a)(d-c)} = \frac{\int\limits_a^b\int\limits_c^d x^2y^2 dxdy}{(b-a)(d-c)} = \frac{\int\limits_a^b x^2 dx}{b-a}\times\frac{\int\limits_c^d y^2 dy}{d-c} = \frac{\frac{x^3}{3}\big|_a^b}{b-a}\times\frac{\frac{y^3}{3}\big|_c^d}{d-c} = \frac{1}{9}\frac{b^3-a^3}{b-a}\frac{d^3-c^3}{d-c} = \frac{((a+b)^2-ab)((c+d)^2-cd)}{9} = \frac{12\cdot 9}{9} = 12$$
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