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Question Number 193513 by Mastermind last updated on 15/Jun/23

$$\mathrm{Evaluate}\mathrm{I}=\underset{\mathrm{s}}{\int }\int {\mathrm{x}}^3\mathrm{dydz}+{\mathrm{x}}^2\mathrm{ydzdx}.$$$$\mathrm{where}\mathrm{S}\mathrm{is}\mathrm{the}\mathrm{closed}\mathrm{surface}\mathrm{consis}-$$$$\mathrm{ting}\mathrm{of}\mathrm{the}\mathrm{cylinder}{\mathrm{x}}^2+{\mathrm{y}}^2={\mathrm{a}}^2,0⩽\mathrm{z}⩽\mathrm{b}$$$$\mathrm{and}\mathrm{the}\mathrm{cylinder}\mathrm{disks}\mathrm{z}=0\mathrm{and}\mathrm{z}=\mathrm{b},$$$${\mathrm{x}}^2+{\mathrm{y}}^2=\mathrm{b},{\mathrm{x}}^2+{\mathrm{y}}^2⩽\mathrm{a}.$$$$$$$$$$$$\mathrm{Help}!$$

Answered by witcher3 last updated on 16/Jun/23

$$(\mathrm{x},\mathrm{y},\mathrm{z})\to (\mathrm{rcos}\left(\mathrm{t}\right),\mathrm{rsin}\left(\mathrm{t}\right),\mathrm{z})$$$$\mathrm{dxdydz}=\mathrm{rdrdtdz}$$

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