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# (Solved): Consider the function $$f$$ and region E. $f(x, y, z)=z ; E=\left\{(x, y, z) \mid x^{2}+y^{2}+z ... Consider the function $$f$$ and region E. \[ f(x, y, z)=z ; E=\left\{(x, y, z) \mid x^{2}+y^{2}+z^{2}-2 z \leq 0, \sqrt{x^{2}+y^{2}} \leq z\right\}$ (a) Express the region $$E$$ in spherical coordinates. $\begin{array}{l} E=\left\{(\rho, \theta, \varphi) \mid 0 \leq \rho \leq 2 \cos (\varphi), 0 \leq \theta \leq 2 \pi, 0 \leq \varphi \leq \frac{\pi}{4}\right\} \\ E=\left\{(\rho, \theta, \varphi) \mid 0 \leq \rho \leq 2 \sin (\varphi), 0 \leq \theta \leq 2 \pi, 0 \leq \varphi \leq \frac{\pi}{4}\right\} \\ E=\left\{(\rho, \theta, \varphi) \mid 0 \leq \rho \leq 2 \sin (\varphi), 0 \leq \theta \leq \pi, 0 \leq \varphi \leq \frac{\pi}{2}\right\} \\ E=\{(\rho, \theta, \varphi) \mid 0 \leq \rho \leq 2 \cos (\varphi), 0 \leq \theta \leq 2 \pi, 0 \leq \varphi \leq \pi\} \\ E=\left\{(\rho, \theta, \varphi) \mid 0 \leq \rho \leq 2 \cos (\varphi), 0 \leq \theta \leq \pi, 0 \leq \varphi \leq \frac{\pi}{2}\right\} \end{array}$ Express the function $$f$$ in spherical coordinates. $f(\rho, \theta, \rho)=$ (b) Convert the integral $$\iiint_{B} f(x, y, z) d V$$ into spherical coordinates and evaluate it.

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### Expert Answer

we have given , f(x,y,z)=z;E:{(x,y,z)?x2+y2+z2?2z?0,x2+y22?z
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