求 $$ex I=iiint_V|x+y+2z|cdot |4x+4y-z| d x d y d z, eex$$ 其中 $V$ 是区域 $dps{x^2+y^2+frac{z^2}{4}leq 1}$.
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 $$eex ea I&=4iiint_{ ilde u^2+ ilde v^2+ ilde w^2leq 1} |3sqrt{2} ilde u|cdot |3 ilde v|
d ilde u
d ilde v
d ilde w\ &=36sqrt{2}iiint_{x^2+y^2+z^2leq 1} |xy|
d x
d y
d z\ &=144sqrt{2}iiint_{x^2+y^2+z^2leq 1atop xgeq 0,ygeq 0} xy
d x
d y
d z\ &=144sqrt{2}int_{-1}^1
d z iint_{x^2+y^2leq 1-z^2atop xgeq0,ygeq 0}xy
d x
d y\ &=144sqrt{2}int_{-1}^1
d z int_0^{sqrt{1-z^2}}
d r int_0^frac{pi}{2} rcos hetacdot rsin hetacdot r
d heta\ &=frac{96sqrt{2}}{5}. eea eeex$$
点击此处查看答案