![11.\; \; V:\{z=0,\; z=4,\; x^2+y^2=4\}\\\\\iiint \limits _{V}(2-xy)dxdydz=\iint \limits _{D_{xy}}(2-xy)dxdy\int \limits ^4_0dz=\\\\=4\iint \limits _{D_{xy}}(2-xy)dxdy=[\, x=rcos\phi ,\; y=rsin\phi ,dxdy=rdrd\phi ,\\\\x^2+y^2=r^2,\; r^2=4\; ,\; r=2\, ]=4\int\limits _0^{2\pi }\, d\phi \int\limits^2_0\, (2-r&^2\, cos\phi \, sin\phi )\, r\, dr=\\\\=4\int\limits^{2\pi }_0\, d\phi \int\limits^2_0(2r-\frac{1}{2}r^3sin2\phi )dr=4\int\limits^{2\pi }_0\, \Big (r^2-\frac{r^4}{8}sin2\phi \Big )\Big |_0^2d\phi =](https://tex.z-dn.net/?f=+11.%5C%3B+%5C%3B+V%3A%5C%7Bz%3D0%2C%5C%3B+z%3D4%2C%5C%3B+x%5E2%2By%5E2%3D4%5C%7D%5C%5C%5C%5C%5Ciiint+%5Climits+_%7BV%7D%282-xy%29dxdydz%3D%5Ciint+%5Climits+_%7BD_%7Bxy%7D%7D%282-xy%29dxdy%5Cint+%5Climits+%5E4_0dz%3D%5C%5C%5C%5C%3D4%5Ciint+%5Climits+_%7BD_%7Bxy%7D%7D%282-xy%29dxdy%3D%5B%5C%2C+x%3Drcos%5Cphi+%2C%5C%3B+y%3Drsin%5Cphi+%2Cdxdy%3Drdrd%5Cphi+%2C%5C%5C%5C%5Cx%5E2%2By%5E2%3Dr%5E2%2C%5C%3B+r%5E2%3D4%5C%3B+%2C%5C%3B+r%3D2%5C%2C+%5D%3D4%5Cint%5Climits+_0%5E%7B2%5Cpi+%7D%5C%2C+d%5Cphi+%5Cint%5Climits%5E2_0%5C%2C+%282-r%26%5E2%5C%2C+cos%5Cphi+%5C%2C+sin%5Cphi+%29%5C%2C+r%5C%2C+dr%3D%5C%5C%5C%5C%3D4%5Cint%5Climits%5E%7B2%5Cpi+%7D_0%5C%2C+d%5Cphi+%5Cint%5Climits%5E2_0%282r-%5Cfrac%7B1%7D%7B2%7Dr%5E3sin2%5Cphi+%29dr%3D4%5Cint%5Climits%5E%7B2%5Cpi+%7D_0%5C%2C+%5CBig+%28r%5E2-%5Cfrac%7Br%5E4%7D%7B8%7Dsin2%5Cphi+%5CBig+%29%5CBig+%7C_0%5E2d%5Cphi+%3D+ "11.\; \; V:\{z=0,\; z=4,\; x^2+y^2=4\}\\\\\iiint \limits _{V}(2-xy)dxdydz=\iint \limits _{D_{xy}}(2-xy)dxdy\int \limits ^4_0dz=\\\\=4\iint \limits _{D_{xy}}(2-xy)dxdy=[\, x=rcos\phi ,\; y=rsin\phi ,dxdy=rdrd\phi ,\\\\x^2+y^2=r^2,\; r^2=4\; ,\; r=2\, ]=4\int\limits _0^{2\pi }\, d\phi \int\limits^2_0\, (2-r&^2\, cos\phi \, sin\phi )\, r\, dr=\\\\=4\int\limits^{2\pi }_0\, d\phi \int\limits^2_0(2r-\frac{1}{2}r^3sin2\phi )dr=4\int\limits^{2\pi }_0\, \Big (r^2-\frac{r^4}{8}sin2\phi \Big )\Big |_0^2d\phi =")
0\; \to \; 0^\circ <(\vec{n}_0,OZ)<90^\circ \\\\\iint \limits_{S}\, z\, dx\, dy=\iint \limits_{D_{xy}}\, (1-x+y)\, dx\, dy=\int\limits^1_0\, dx \int\limits_{x-1}^0\, (1-x+y)\, dy=\\\\=\int\limits^1_0\Big ((1-x)\cdot y+\frac{y^2}{2}\Big )\Big |^0_{x-1}\, dx=\int\limits^1_0\, \Big ((x-1)^2-\frac{(x-1)^2}{2}\Big )\, dx=\\\\=\int\limits^1_0\, \frac{(x-1)^2}{2}\, dx=\frac{(x-1)^3}{6}\Big |_0^1=\frac{1}{6}\cdot (0-(-1))=\frac{1}{6}" alt="12.\; \; S:\; \; x-y+z=1\; \to \; \; \vec{n}=(1,-1,1)\; ,\; \vec{n}_0=(\frac{1}{\sqrt3},-\frac{1}{\sqrt3},\frac{1}{\sqrt3})\\\\cos(OZ,\vec{n}_0)>0\; \to \; 0^\circ <(\vec{n}_0,OZ)<90^\circ \\\\\iint \limits_{S}\, z\, dx\, dy=\iint \limits_{D_{xy}}\, (1-x+y)\, dx\, dy=\int\limits^1_0\, dx \int\limits_{x-1}^0\, (1-x+y)\, dy=\\\\=\int\limits^1_0\Big ((1-x)\cdot y+\frac{y^2}{2}\Big )\Big |^0_{x-1}\, dx=\int\limits^1_0\, \Big ((x-1)^2-\frac{(x-1)^2}{2}\Big )\, dx=\\\\=\int\limits^1_0\, \frac{(x-1)^2}{2}\, dx=\frac{(x-1)^3}{6}\Big |_0^1=\frac{1}{6}\cdot (0-(-1))=\frac{1}{6}" align="absmiddle" class="latex-formula">