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(Solved): Consider the function \( f \) and region \( E \). \[ f(x, y, z)=y, E=\left\{(x, y, z) \mid 1 \leq x ...




Consider the function \( f \) and region \( E \).
\[
f(x, y, z)=y, E=\left\{(x, y, z) \mid 1 \leq x^{2}+z^{2} \leq 9,0 \leq y
Consider the function \( f \) and region \( E \). \[ f(x, y, z)=y, E=\left\{(x, y, z) \mid 1 \leq x^{2}+z^{2} \leq 9,0 \leq y \leq 1-x^{2}-z^{2}\right\} \] (a) Express the region \( E \) in cylindrical coordinates. \[ \begin{aligned} E &=\left\{(r, \theta, z) \mid 1 \leq r \leq 3,0 \leq \theta \leq \pi, 0 \leq z \leq 1-r^{2}\right\} \\ E &=\{(r, \theta, z) \mid 1 \leq r \leq 9,0 \leq \theta \leq 2 \pi, 0 \leq z \leq 1-\\ \left.r^{2}\right\} & \\ E &=\{(r, \theta, z) \mid 1 \leq r \leq 9,0 \leq \theta \leq \pi, 0 \leq z \leq 1-r\} \\ & E=\{(r, \theta, z) \mid 1 \leq r \leq 3,0 \leq \theta \leq 2 \pi, 0 \leq z \leq 1-\\ \left.r^{2}\right\} & \\ E &=\{(r, \theta, z) \mid 1 \leq r \leq 3,0 \leq \theta \leq 2 \pi, 0 \leq z \leq 1-r\} \end{aligned} \] Express the function \( f \) in cylindrical coordinates. \[ f(r, \theta, z)= \] (b) Convert the integral \( \iiint_{E} f(x, y, z) d V \) into cylindrical coordinates and evaluate it.


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