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Question Number 134122 by Ahmed1hamouda last updated on 28/Feb/21

Answered by TheSupreme last updated on 28/Feb/21

$$x=1+\rho cos\left(\theta \right)$$$$y=\rho sin\left(\theta \right)$$$$z=z$$$$det\left(J\right)=\rho$$$$E=\{(\rho ,\theta ,z)\mid 0<\theta <2\pi ,0<\rho <1,0<z<\rho sin\theta \}$$$${\int }_0^{2\pi }{\int }_0^1{\int }_0^{\rho sin\theta }\rho dzd\rho d\theta =$$$$2{\int }_0^{\pi }{\int }_0^1{\rho }^{2}sin\theta d\theta d\rho =$$$$2{[-cos\theta ]}_0^{\pi }{\left[\frac{{\rho }^3}{3}\right]}_0^1=\frac{4}{3}$$

Commented by Ahmed1hamouda last updated on 28/Feb/21

$${\int }_0^{2\pi }{\int }_0^1{\rho }^{2}sin\theta drd\theta =0\cancel{=}$$

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