${\int }_{-\frac{2}{3}\sqrt{5}}^2{\int }_{\frac{3}{4}\left(\sqrt{5}-3\right) \left( y-2\right)}^{\frac{3}{2}\sqrt{4-y^2}}\left(1+2 x^2+3 y^2\right) \mathit{\ⅆ}x \mathit{\ⅆ}y$

= $51\arcsin \left(\frac{\sqrt{3}}{6}\left(1\+ \sqrt{5}\right)\right)\+\frac{7}{3}\left(\sqrt{5}-\frac{14}{3}\right)$$$

$\doteq 55.83$

Initialize

Loading Student:-MultivariateCalculus

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

$\mathrm{XL}\=\frac{3}{4}\left(\sqrt{5}-3\right) \left( y-2\right)$$\stackrel{\text{assign}}{\to }$$$

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

$\mathrm{YB}\=-2\sqrt{5}\/3$$\stackrel{\text{assign}}{\to }$

Access the MultiInt command via the Context Panel

$f$ = $2x^2\+3y^2\+1$$\stackrel{\text{MultiInt}}{\to }$${\∫}_{-\frac{2}{3}\sqrt{5}}^2{\∫}_{\frac{3}{4}\left(\sqrt{5}-3\right)\left(y-2\right)}^{\frac{3}{2}\sqrt{-y^2\+4}}\left(2x^2\+3y^2\+1\right)\ⅆx\ⅆy$$\stackrel{\text{at 10 digits}}{\to }$$55.83116045$$$

Tools≻Tasks≻Browse:

Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻

Table 5.3.9(b)   Visualizing $R$ and the resulting volume for iteration in the order $\mathrm{dy} \mathrm{dx}$ 

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

${\int }_{\mathrm{YB}}^2{\int }_{\mathrm{XL}}^{\mathrm{XR}}f \mathit{\ⅆ}x \mathit{\ⅆ}y$ = $51\arcsin \left(\frac{1}{6}\sqrt{3}\+\frac{1}{6}\sqrt{5}\sqrt{3}\right)-\frac{98}{9}\+\frac{7}{3}\sqrt{5}$$\stackrel{\text{at 10 digits}}{\to }$$55.83116051$

Display the iterated integrals

${\int }_{\mathrm{YB}}^2{\int }_{\mathrm{XL}}^{\mathrm{XR}}f \mathit{\ⅆ}x \mathit{\ⅆ}y$

${\∫}_{-\frac{2}{3}\sqrt{5}}^2{\∫}_{\frac{3}{4}\left(\sqrt{5}-3\right)\left(y-2\right)}^{\frac{3}{2}\sqrt{-y^2\+4}}\left(2x^2\+3y^2\+1\right)\ⅆx\ⅆy$

$\=$

$51\arcsin \left(\frac{1}{6}\sqrt{3}\+\frac{1}{6}\sqrt{5}\sqrt{3}\right)-\frac{98}{9}\+\frac{7}{3}\sqrt{5}$

$y\=2-\left(1\+\sqrt{5}\/3\right) x$$\stackrel{\text{solutions for x}}{\to }$$-\frac{3\left(y-2\right)}{3\+\sqrt{5}}$$\stackrel{\text{rationalize}}{\=}$$\frac{3}{4}\left(\sqrt{5}-3\right)\left(y-2\right)$

Initialize

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

$f≔1\+2 x^2\+3 y^2\:$

$\mathrm{XL}≔\frac{3}{4}\left(\sqrt{5}-3\right) \left( y-2\right)\:$

$\mathrm{XR}≔3\sqrt{4-y^2}\/2\:$

$\mathrm{YB}≔-2\sqrt{5}\/3\:$

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

$$\begin{gathered} \mathrm{Int}\left(f\,\left[x\=\mathrm{XL}..\mathrm{XR}\,y\=\mathrm{YB}..2\right]\right)\; \\ \mathrm{int}\left(f\,\left[x\=\mathrm{XL}..\mathrm{XR}\,y\=\mathrm{YB}..2\right]\right) \end{gathered}$$

${\∫}_{-\frac{2}{3}\sqrt{5}}^2{\∫}_{\frac{3}{4}\left(\sqrt{5}-3\right)\left(y-2\right)}^{\frac{3}{2}\sqrt{-y^2\+4}}\left(2x^2\+3y^2\+1\right)\ⅆx\ⅆy$

Use the MultiInt command from the Student MultivariateCalculus package

$\mathrm{MultiInt}\left(f\,x\=\mathrm{XL}..\mathrm{XR}\,y\=\mathrm{YB}..2\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_{-\frac{2}{3}\sqrt{5}}^2{\∫}_{\frac{3}{4}\left(\sqrt{5}-3\right)\left(y-2\right)}^{\frac{3}{2}\sqrt{-y^2\+4}}\left(2x^2\+3y^2\+1\right)\ⅆx\ⅆy$

$\mathrm{MultiInt}\left(f\,x\=\mathrm{XL}..\mathrm{XR}\,y\=\mathrm{YB}..2\right)$

$51\arcsin \left(\frac{1}{6}\sqrt{3}\+\frac{1}{6}\sqrt{5}\sqrt{3}\right)-\frac{98}{9}\+\frac{7}{3}\sqrt{5}$