∬Rx2y2+cos(πx)+sin(πy)dA∬Rx2y2+cos(πx)+sin(πy)dA \displaystyle \iint\limits_{R}{{{x^2}{y^2} + \cos \left( {\pi x} \right) + \sin \left( {\pi y} \right)\,dA}}, R=[−2,−1]×[0,1]
My question is: since the integrand is already separated into terms that depend only on (x), only on (y), and both variables, how do you quickly decide whether to integrate with respect to (y) first or (x) first? Is there a general rule of thumb for spotting when one order will make the work noticeably easier?