Calculate the following:
$$\int_S z \, dS$$
where S is the surface $$ z = x^2 + y^2 , x^2 + y^2 \le 1$$
2026-04-01 05:41:47.1775022107
Help on this surface integral?
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It is a paraboloide of revolution defined in cylindrical coordinates by $$r=\sqrt {x^2+y^2}=\sqrt {z} $$ with $$0\le z \le 1.$$
the element of surface is $$2\pi r dz =2\pi\sqrt {z}dz $$
the integrale is $$2\pi\int_0^1z\sqrt {z}dz=$$ $$2\pi\frac {2}{5}=\frac {4\pi}{5} $$