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• Notes

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 $z=4$

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

$r=4$

$\theta=0 \rightarrow 2 \pi$

$z=-5 \rightarrow 4$

$r=0 \rightarrow r=4$

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

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

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

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

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

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

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