${\int }_{-\sqrt{\frac{5}{3}}}^{\sqrt{\frac{5}{3}}}{\int }_{-\sqrt{\frac{25-15 y^2}{6}}}^{\sqrt{\frac{25-15 y^2}{6}}}{\int }_{2 x^2\+5 y^2}^{25-4 x^2-10 y^2}1 \mathrm{dz} \mathrm{dx} \mathrm{dy}$ = $\frac{125}{12}\sqrt{10}\mathrm{\π}$ 

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Loading Student:-MultivariateCalculus

Access the MultiInt command via the Context Panel

$1$$\stackrel{\text{MultiInt}}{\to }$${\∫}_{-\frac{1}{3}\sqrt{15}}^{\frac{1}{3}\sqrt{15}}{\∫}_{-\frac{1}{6}\sqrt{-90y^2\+150}}^{\frac{1}{6}\sqrt{-90y^2\+150}}{\∫}_{2x^2\+5y^2}^{-4x^2-10y^2\+25}1\ⅆz\ⅆx\ⅆy$$\=$$\frac{125}{12}\sqrt{2}\sqrt{5}\mathrm{\π}$

$\sqrt{\left(25-15 y^2\right)\/6}$$\stackrel{\text{assign to a name}}{\to }$$X$

Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Cartesian 3-D

Table 8.1.11(a)   Task template integrating in Cartesian coordinates

${\int }_{-\sqrt{\frac{5}{3}}}^{\sqrt{\frac{5}{3}}}{\int }_{-\sqrt{\frac{25-15 y^2}{6}}}^{\sqrt{\frac{25-15 y^2}{6}}}{\int }_{2 x^2\+5 y^2}^{25-4 x^2-10 y^2}1 \mathit{\ⅆ}z \mathit{\ⅆ}x \mathit{\ⅆ}y$

$\frac{125}{12}\sqrt{2}\sqrt{5}\mathrm{\π}$

$\stackrel{\text{at 10 digits}}{\to }$

$103.4852944$

Table 8.1.11(b)   Integration via first principles

$\mathrm{ZB}≔2 x^2\+5 y^2\:$

$\mathrm{ZT}≔25-4 x^2-10 y^2\:$

$X≔\sqrt{\left(25-15 y^2\right)\/6}\:$

$\mathrm{MultiInt}\left(1\,z\=\mathrm{ZB}..\mathrm{ZT}\,x\=-X..X\,y\=-\sqrt{5\/3}..\sqrt{5\/3}\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_{-\frac{1}{3}\sqrt{15}}^{\frac{1}{3}\sqrt{15}}{\∫}_{-\frac{1}{6}\sqrt{-90y^2\+150}}^{\frac{1}{6}\sqrt{-90y^2\+150}}{\∫}_{2x^2\+5y^2}^{-4x^2-10y^2\+25}1\ⅆz\ⅆx\ⅆy$

$\mathrm{MultiInt}\left(1\,z\=\mathrm{ZB}..\mathrm{ZT}\,x\=-X..X\,y\=-\sqrt{5\/3}..\sqrt{5\/3}\right)$ = $\frac{125}{12}\sqrt{5}\sqrt{2}\mathrm{\π}$

Table 8.1.11(c)   MultiInt command iterating in Cartesian coordinates in the order $\mathrm{dy} \mathrm{dz} \mathrm{dx}$ 

$$\begin{gathered} \mathrm{Int}\left(1\,\left[z\=\mathrm{ZB}..\mathrm{ZT}\,x\=-X..X\,y\=-\sqrt{5\/3}..\sqrt{5\/3}\right]\right)\= \\ \mathrm{int}\left(1\,\left[z\=\mathrm{ZB}..\mathrm{ZT}\,x\=-X..X\,y\=-\sqrt{5\/3}..\sqrt{5\/3}\right]\right) \end{gathered}$$

${\∫}_{-\frac{1}{3}\sqrt{15}}^{\frac{1}{3}\sqrt{15}}{\∫}_{-\frac{1}{6}\sqrt{-90y^2\+150}}^{\frac{1}{6}\sqrt{-90y^2\+150}}{\∫}_{2x^2\+5y^2}^{-4x^2-10y^2\+25}1\ⅆz\ⅆx\ⅆy\=\frac{125}{12}\sqrt{2}\sqrt{5}\mathrm{\π}$

Table 8.1.11(d)   Top-level Int and int commands