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Loading Student:-MultivariateCalculus

$f\=1\+2 x^2\+3 y^2$$\stackrel{\text{assign}}{\to }$

$\mathrm{YT}\=2\sqrt{9-x^2}\/3$$\stackrel{\text{assign}}{\to }$

$\mathrm{YB}\=2-2 x\/3$$\stackrel{\text{assign}}{\to }$

$\mathrm{XR}\=3\sqrt{4-y^2}\/2$$\stackrel{\text{assign}}{\to }$

$\mathrm{XL}\=3-3 y\/2$$\stackrel{\text{assign}}{\to }$

Access the MultiInt command via the Context Panel

$f$ = $2x^2\+3y^2\+1$$\stackrel{\text{MultiInt}}{\to }$${\∫}_0^3{\∫}_{2-\frac{2}{3}x}^{\frac{2}{3}\sqrt{-x^2\+9}}\left(2x^2\+3y^2\+1\right)\ⅆy\ⅆx$$\=$$-18\+\frac{51}{4}\mathrm{\π}$ 

Tools≻Tasks≻Browse:

Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻

Table 5.3.8(a)   Visualizing $R$ and the resulting volume for iteration in the order $\mathrm{dy} \mathrm{dx}$ 

Tools≻Tasks≻Browse:

Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻

Table 5.3.8(b)   Visualizing $R$ and the resulting volume for iteration in the order $\mathrm{dx} \mathrm{dy}$ 

Iterate $\mathrm{dy} \mathrm{dx}$ and $\mathrm{dx} \mathrm{dy}$ 

${\int }_0^3{\int }_{2-2 x\/3}^{\frac{2}{3}\sqrt{9-x^2}}f \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $-18\+\frac{51}{4}\mathrm{\π}$$$

${\int }_0^2{\int }_{3-3 y\/2}^{\frac{3}{2}\sqrt{4-y^2}}f \mathit{\ⅆ}x \mathit{\ⅆ}y$ = $-18\+\frac{51}{4}\mathrm{\π}$$$

Display the iterated integrals

${\int }_0^3{\int }_{2-2 x\/3}^{\frac{2}{3}\sqrt{9-x^2}}f \mathit{\ⅆ}y \mathit{\ⅆ}x$

${\∫}_0^3{\∫}_{2-\frac{2}{3}x}^{\frac{2}{3}\sqrt{-x^2\+9}}\left(2x^2\+3y^2\+1\right)\ⅆy\ⅆx$

$\=$

$-18\+\frac{51}{4}\mathrm{\π}$

${\int }_0^2{\int }_{3-3 y\/2}^{\frac{3}{2}\sqrt{4-y^2}}f \mathit{\ⅆ}x \mathit{\ⅆ}y$

${\∫}_0^2{\∫}_{3-\frac{3}{2}y}^{\frac{3}{2}\sqrt{-y^2\+4}}\left(2x^2\+3y^2\+1\right)\ⅆx\ⅆy$

$\=$

$-18\+\frac{51}{4}\mathrm{\π}$

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$\mathrm{with}\left(\mathrm{Student}:-\mathrm{MultivariateCalculus}\right)\:$

$f≔1\+2 x^2\+3 y^2\:$

Iterate in the order $\mathrm{dy} \mathrm{dx}$

$\mathrm{Int}\left(f\,\left[y\=2-\frac{2}{3}x..\frac{2}{3} \sqrt{9-x^2}\,x\=0..3\right]\right)\=\mathrm{int}\left(f\,\left[y\=2-\frac{2}{3}x..\frac{2}{3} \sqrt{9-x^2}\,x\=0..3\right]\right)$

${\∫}_0^3{\∫}_{2-\frac{2}{3}x}^{\frac{2}{3}\sqrt{-x^2\+9}}\left(2x^2\+3y^2\+1\right)\ⅆy\ⅆx\=-18\+\frac{51}{4}\mathrm{\π}$

Iterate in the order $\mathrm{dx} \mathrm{dy}$ 

$\mathrm{Int}\left(f\,\left[x\=3-\frac{3}{2}y..\frac{3}{2}\sqrt{4-y^2}\,y\=0..2\right]\right)\=\mathrm{int}\left(f\,\left[x\=3-\frac{3}{2}y..\frac{3}{2}\sqrt{4-y^2}\,y\=0..2\right]\right)$

${\∫}_0^2{\∫}_{3-\frac{3}{2}y}^{\frac{3}{2}\sqrt{-y^2\+4}}\left(2x^2\+3y^2\+1\right)\ⅆx\ⅆy\=-18\+\frac{51}{4}\mathrm{\π}$

Use the MultiInt command from the Student MultivariateCalculus package

$\mathrm{MultiInt}\left(f\,y\=2-\frac{2}{3}x..\frac{2}{3} \sqrt{9-x^2}\,x\=0..3\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^3{\∫}_{2-\frac{2}{3}x}^{\frac{2}{3}\sqrt{-x^2\+9}}\left(2x^2\+3y^2\+1\right)\ⅆy\ⅆx$

$\mathrm{MultiInt}\left(f\,y\=2-\frac{2}{3}x..\frac{2}{3} \sqrt{9-x^2}\,x\=0..3\right)$

$-18\+\frac{51}{4}\mathrm{\π}$

$\mathrm{MultiInt}\left(f\,x\=3-\frac{3}{2}y..\frac{3}{2}\sqrt{4-y^2}\,y\=0..2\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^2{\∫}_{3-\frac{3}{2}y}^{\frac{3}{2}\sqrt{-y^2\+4}}\left(2x^2\+3y^2\+1\right)\ⅆx\ⅆy$

$\mathrm{MultiInt}\left(f\,x\=3-\frac{3}{2}y..\frac{3}{2}\sqrt{4-y^2}\,y\=0..2\right)$

$-18\+\frac{51}{4}\mathrm{\π}$

Obtain stepwise evaluations via the MultiInt command

$\mathrm{MultiInt}\left(f\,y\=2-\frac{2}{3}x..\frac{2}{3} \sqrt{9-x^2}\,x\=0..3\,\mathrm{output}\=\mathrm{steps}\right)\:$

${\int }_0^3{\int }_{2-\frac{2}{3}x}^{\frac{2}{3}\sqrt{-x^2\+9}}\left(2x^2\+3y^2\+1\right)\ⅆy\ⅆx$

                $\=\frac{51}{4}\mathrm{\π}-18$

$\mathrm{MultiInt}\left(f\,x\=3-\frac{3}{2}y..\frac{3}{2}\sqrt{4-y^2}\,y\=0..2\,\mathrm{output}\=\mathrm{steps}\right)\:$

${\int }_0^2{\int }_{3-\frac{3}{2}y}^{\frac{3}{2}\sqrt{-y^2\+4}}\left(2x^2\+3y^2\+1\right)\ⅆx\ⅆy$

               $\=\frac{51}{4}\mathrm{\π}-18$