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Evaluate-I-s-x-3-dydz-x-2-ydzdx-where-S-is-the-closed-surface-consis-ting-of-the-cylinder-x-2-y-2-a-2-0-z-b-and-the-cylinder-disks-z-0-and-z-b-x-2-y-2-b-x-2-y-2-a-Help-




Question Number 193513 by Mastermind last updated on 15/Jun/23
Evaluate I=∫_s ∫x^3 dydz + x^2 ydzdx.  where S is the closed surface consis−  ting of the cylinder x^2 +y^2 =a^2 , 0≤z≤b  and the cylinder  disks z=0 and z=b,  x^2 +y^2 =b, x^2 +y^2 ≤a.      Help!
$$\mathrm{Evaluate}\:\mathrm{I}=\underset{\mathrm{s}} {\int}\int\mathrm{x}^{\mathrm{3}} \mathrm{dydz}\:+\:\mathrm{x}^{\mathrm{2}} \mathrm{ydzdx}. \\ $$$$\mathrm{where}\:\mathrm{S}\:\mathrm{is}\:\mathrm{the}\:\mathrm{closed}\:\mathrm{surface}\:\mathrm{consis}− \\ $$$$\mathrm{ting}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cylinder}\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} =\mathrm{a}^{\mathrm{2}} ,\:\mathrm{0}\leqslant\mathrm{z}\leqslant\mathrm{b} \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{cylinder}\:\:\mathrm{disks}\:\mathrm{z}=\mathrm{0}\:\mathrm{and}\:\mathrm{z}=\mathrm{b}, \\ $$$$\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} =\mathrm{b},\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \leqslant\mathrm{a}. \\ $$$$ \\ $$$$ \\ $$$$\mathrm{Help}! \\ $$
Answered by witcher3 last updated on 16/Jun/23
(x,y,z)→(rcos(t),rsin(t),z)  dxdydz=rdrdtdz
$$\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)\rightarrow\left(\mathrm{rcos}\left(\mathrm{t}\right),\mathrm{rsin}\left(\mathrm{t}\right),\mathrm{z}\right) \\ $$$$\mathrm{dxdydz}=\mathrm{rdrdtdz} \\ $$

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