${\int }_0^5{\int }_0^{3\left(1-\frac{x}{5}\right)}{\int }_0^{\frac{\left(15-3 x-5 y\right)}{7}}1 \mathrm{dz} \mathrm{dy} \mathrm{dx}$ = $\frac{75}{14}$

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Calculus - Multivariate≻Integration≻Multiple Integration≻Cartesian 3-D

Table 8.1.2(a)   Task template implementing the MultiInt command iterating in the order $\mathrm{dz} \mathrm{dy} \mathrm{dx}$

Initialize

Loading Student:-MultivariateCalculus

Access the MultiInt command via the Context Panel

$1$$\stackrel{\text{MultiInt}}{\to }$${\∫}_0^5{\∫}_0^{3-\frac{3}{5}x}{\∫}_0^{\frac{15}{7}-\frac{3}{7}x-\frac{5}{7}y}1\ⅆz\ⅆy\ⅆx$$\=$$\frac{75}{14}$

${\int }_0^5{\int }_0^{3\left(1-x\/5\right)}{\int }_0^{\left(15-3 x-5 y\right)\/7}1 \mathit{\ⅆ}z \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $\frac{75}{14}$$$

Table 8.1.2(b)   Integration via first principles

$$\begin{gathered} V≔⟨0\,0\,0⟩\,⟨5\,0\,0⟩\,⟨0\,3\,0⟩\,⟨0\,0\,15\/7⟩\: \\ \mathrm{MultiInt}\left(1\,\left[x\,y\,z\right]\=\mathrm{Tetrahedron}\left(V\right)\,\mathrm{output}\=\mathrm{integral}\right) \end{gathered}$$

$\frac{15}{7}{\∫}_0^1{\∫}_0^{5-5t}{\∫}_0^{\frac{\left(-3\+3t\right)\left(x-5\+5t\right)}{5-5t}}1\ⅆy\ⅆx\ⅆt$

$\mathrm{MultiInt}\left(1\,\left[x\,y\,z\right]\=\mathrm{Tetrahedron}\left(V\right)\right)$ = $\frac{75}{14}$$$

Table 8.1.2(c)   Integration over a tetrahedron

$\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{MultiInt}\left(1\,z\=0..\left(15-3 x-5 y\right)\/7\,y\=0..3\left(1-x\/5\right)\,x\=0..5\right)$

$\frac{75}{14}$

Table 8.1.2(d)   MultiInt command iterating in the order $\mathrm{dz} \mathrm{dy} \mathrm{dx}$

$$\begin{gathered} \mathrm{Int}\left(1\,\left[z\=0..\left(15-3 x-5 y\right)\/7\,y\=0..3\left(1-x\/5\right)\,x\=0..5\right]\right)\= \\ \mathrm{int}\left(1\,\left[z\=0..\left(15-3 x-5 y\right)\/7\,y\=0..3\left(1-x\/5\right)\,x\=0..5\right]\right) \end{gathered}$$

${\∫}_0^5{\∫}_0^{3-\frac{3}{5}x}{\∫}_0^{\frac{15}{7}-\frac{3}{7}x-\frac{5}{7}y}1\ⅆz\ⅆy\ⅆx\=\frac{75}{14}$

Table 8.1.2(e)   Top-level Int and int commands