[家里蹲大学数学杂志]第241期利用正交变换和对称性求解三重积分

解答: 作变换 $$ex x=u,quad y=v,quad frac{z}{2}=w, eex$$ 则 $$eex ea I&=iiint_{u^2+v^2+w^2leq 1} |u+v+4w|cdot |4u+4v-2w|cdot 2 d u d v d w\ &=4iiint_{u^2+v^2+w^2leq 1}|u+v+4w|cdot |2u+2v-w| d u d v d w. eea eeex$$ 再作变换 $$ex ilde u=frac{u+v+4w}{3sqrt{2}},quad ilde v=frac{2u+2v-w}{3},quad ilde w=frac{-u+v}{sqrt{2}}, eex$$ 则 $$eex ea I&=4iiint_{ ilde u^2+ ilde v^2+ ilde w^2leq 1} |3sqrt{2} ilde u|cdot |3 ilde v| d ilde u d ilde v d ilde w\ &=36sqrt{2}iiint_{x^2+y^2+z^2leq 1} |xy| d x d y d z\ &=144sqrt{2}iiint_{x^2+y^2+z^2leq 1atop xgeq 0,ygeq 0} xy d x d y d z\ &=144sqrt{2}int_{-1}^1 d z iint_{x^2+y^2leq 1-z^2atop xgeq0,ygeq 0}xy d x d y\ &=144sqrt{2}int_{-1}^1 d z int_0^{sqrt{1-z^2}} d r int_0^frac{pi}{2} rcos hetacdot rsin hetacdot r d heta\ &=frac{96sqrt{2}}{5}. eea eeex$$