<prompt> Acha mathcalSkushotox, y, z katika mathbbR3 mid x2y2z2 1kulia inaashiria nyanja ya kitengo katika mathbbR3. Tathmini uunganisho wa uso zaidi ya mathcalS :iint_mathcalSleftx2yzkulia d A .  <response>: Kwa kutumia mabadiliko ya vigezoanzakesixsin varphi cos theta 0theta2 pi ysin varphi sin theta 0varphipi zcos varphi mwishokesitunad Asin varphi d theta d varphinaiint_mathcalSkushotox2yzkulia d Aint_0pi int_02 pikushoto sin 2 varphi cos 2 thetasin varphi sin thetacos varphiright sin varphi d theta d varphi .Kuvunja muhimu kwa maneno matatu, tunapataanzailiyopangwaint_0pi int_02 pi sin varphi cos varphi d theta d varphi2 pi cdot frac12 int_0pi sin 2 varphi d varphi0 nint_0pi int_02 pi sin 2 varphi sin theta d theta d varphileftint_0pi sin 2 varphi d varphiright int_02 pi sin theta d theta0 nint_0pi int_02 pi sin 3 varphi cos 2 theta d theta d varphi leftint_0pi sin 3 varphi d varphirightleftint_02 pi cos 2 theta d thetakulia nint_0pi frac14kushoto3 sin varphi-sin 3 varphikulia d varphi int_02 pi cos 2 theta d theta nfrac14left-3 cos varphi kushoto.frac13 cos 3 varphikulia_0 pikuliakushotoint_02 pi frac 1cos 2 theta2 d thetakulia nfrac14left-3-2frac13-- 2kulia pi nfrac14left6-frac23right pifrac43 pi .mwishoiliyopangwaKwa hiyo,iint_mathcalSkushotox2yzkulia d Afrac43 pi 