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Question Number 169706 by MikeH last updated on 06/May/22
using cylindrical coordinates  { ((x=rcosθ)),((y = rsin θ)),((z=z)) :}  to evaluate the integral  K= ∫∫∫_S (√(x^2 +y^2 −z^2 )) dxdydz  where   S= {(x,y,z) ∈R^3 : x^2 +y^2  ≤ 4, 0 ≤z≤(√(x^2 +y^2 ))}
$$\mathrm{using}\:\mathrm{cylindrical}\:\mathrm{coordinates}\:\begin{cases}{{x}={r}\mathrm{cos}\theta}\\{{y}\:=\:{r}\mathrm{sin}\:\theta}\\{{z}={z}}\end{cases} \\ $$$$\mathrm{to}\:\mathrm{evaluate}\:\mathrm{the}\:\mathrm{integral} \\ $$$${K}=\:\int\int\int_{{S}} \sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} −{z}^{\mathrm{2}} }\:{dxdydz} \\ $$$$\mathrm{where} \\ $$$$\:{S}=\:\left\{\left({x},{y},{z}\right)\:\in\mathbb{R}^{\mathrm{3}} :\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \:\leqslant\:\mathrm{4},\:\mathrm{0}\:\leqslant{z}\leqslant\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }\right\} \\ $$

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