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Use cylindrical coordinates to Evaluate $$\iiint_{E} \sqrt{x^{2}+y^{2}} d V$$
where $$E$$ is the region that lies inside the cylinder
$$x^{2}+y^{2}=16$$ and between the planes $$z=-5$$ and $$z=4$$

$$x^{2}+y^{2}=16=r^{2}$$

$$r=4$$

$$\theta=0 \rightarrow 2 \pi$$

$$z=-5 \rightarrow 4$$

$$r=0 \rightarrow r=4$$

$$\iiint \sqrt{x^{2}+y^{2}}=\sqrt {r^{2}} $$

$$d v=d x d y d z$$
$$d v=r d \theta d r dz$$

$$dx dy dz \rightarrow r d \theta d r d z$$

$$\int_{-4}^{5} \int_{0}^{2 \pi} \int_{0}^{4} r \cdot r d r d \theta d z \Rightarrow r^{2} $$

$$=\left.\int_{-4}^{5} \int_{0}^{2 \pi} r^{3} / 3\right|_{0} ^{4} d \theta d z$$

$$=\int_{0}^{2 \pi}\left(\frac{4^{3}}{3}\right) z|_{-4}^{5} d \theta=\frac{4^{3}}{3} * 9 * 2 \pi=384 \pi$$

$$\int d \theta = \theta  |_{0}^{2 \pi} $$

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