# Evaluate the iterated integral: a)int_0^3 int_0^(pi/3) int_0^4 r xx cos theta drd theta dz b)int_0^(pi/2) int_0^3 int_0^(4 - z) zdrdzd theta

Evaluate the iterated integral:
a) ${\int }_{0}^{3}{\int }_{0}^{\frac{\pi }{3}}{\int }_{0}^{4}r×\mathrm{cos}\theta drd\theta dz$

b) ${\int }_{0}^{\frac{\pi }{2}}{\int }_{0}^{3}{\int }_{0}^{4-z}zdrdzd\theta$

c) ${\int }_{0}^{\frac{\pi }{2}}{\int }_{0}^{\frac{\pi }{2}}{\int }_{0}^{2}{\rho }^{2}d\rho d\theta d\varphi$

d) ${\int }_{0}^{\frac{\pi }{4}}{\int }_{0}^{\frac{\pi }{4}}{\int }_{0}^{\mathrm{cos}\left(\varphi \right)}\mathrm{cos}\theta d\rho d\varphi d\theta$

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2abehn

a) ${\int }_{0}^{3}{\int }_{0}^{\frac{\pi }{3}}{\int }_{0}^{4}r×\mathrm{cos}\theta drd\theta dz=$

$={\int }_{0}^{3}{\int }_{0}^{\frac{\pi }{3}}\mathrm{cos}\theta {\left[{\gamma }^{\frac{2}{2}}\right]}_{0}^{4}d\theta dz=$

$={\int }_{0}^{3}{\int }_{0}^{\frac{\pi }{3}}8\mathrm{cos}\theta ×d\theta ×dz=$

$={\int }_{0}^{3}8×{\left[\mathrm{sin}\theta \right]}_{0}^{\frac{\pi }{3}}dz=$

$={\int }_{0}^{3}8×\frac{\sqrt{3}}{2}dz=$

$=4\sqrt{3}{\int }_{0}^{3}dz=$

$=4\sqrt{3}{\left[z\right]}_{0}^{3}=$

$=12\sqrt{3}$

b) ${\int }_{0}^{\frac{\pi }{2}}{\int }_{0}^{3}{\int }_{0}^{4-z}zdrdzd\theta =$

$={\int }_{0}^{\frac{\pi }{2}}{\int }_{0}^{3}z×{\left[\gamma \right]}_{0}^{4-z}dz×d\theta =$

$={\int }_{0}^{\frac{\pi }{2}}{\int }_{0}^{3}z×\left[\left(4–2\right)–0\right]dzd\theta =$

$={\int }_{0}^{\frac{\pi }{2}}{\int }_{0}^{3}4z–{z}^{2}×dzd\theta =$

$={\int }_{0}^{\frac{\pi }{2}}{\left[\left(4\frac{{z}^{2}}{2}\right)–\frac{{z}^{3}}{3}\right]}_{0}^{3}d\theta =$

$={\int }_{0}^{\frac{\pi }{2}}\left(18-9\right)×d\theta =$

$=9{\int }_{0}^{\frac{\pi }{2}}d\theta =$

$=9{\left[\theta \right]}_{0}^{\frac{\pi }{2}}=$

$=\frac{9\pi }{2}$

c)