Initialize

Loading Student:-MultivariateCalculus

$f\=1\+2 x^2\+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^{\mathrm{\π}}{\∫}_{\cos \left(x\right)}^{1\+\sin \left(x\right)}\left(2x^2\+3y^2\+1\right)\ⅆy\ⅆx$$\=$$\frac{4}{3}\+\frac{2}{3}{\mathrm{\π}}^3\+2{\mathrm{\π}}^2\+\frac{15}{2}\mathrm{\π}$$$

Tools≻Tasks≻Browse:

Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻

Table 5.3.10(a)   Visualizing $R$ and the resulting volume for iteration in the order $\mathrm{dy} \mathrm{dx}$ 

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

${\int }_0^{\mathrm{\π}}{\int }_{\cos \left(x\right)}^{1\+\sin \left(x\right)}f \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $\frac{4}{3}\+\frac{2}{3}{\mathrm{\π}}^3\+2{\mathrm{\π}}^2\+\frac{15}{2}\mathrm{\π}$$$

Display the iterated integrals

${\int }_0^{\mathrm{\pi }}{\int }_{\cos \left(x\right)}^{1\+\sin \left(x\right)}f \mathit{\ⅆ}y \mathit{\ⅆ}x$

${\∫}_0^{\mathrm{\π}}{\∫}_{\cos \left(x\right)}^{1\+\sin \left(x\right)}\left(2x^2\+3y^2\+1\right)\ⅆy\ⅆx$

$\=$

$\frac{4}{3}\+\frac{2}{3}{\mathrm{\π}}^3\+2{\mathrm{\π}}^2\+\frac{15}{2}\mathrm{\π}$

Initialize

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

$f≔1\+2 x^2\+3 y^2\:$

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

$\mathrm{Int}\left(f\,\left[y\=\cos \left(x\right)..1\+\sin \left(x\right)\,x\=0..\mathrm{\π}\right]\right)\=\mathrm{int}\left(f\,\left[y\=\cos \left(x\right)..1\+\sin \left(x\right)\,x\=0..\mathrm{\π}\right]\right)$

${\∫}_0^{\mathrm{\π}}{\∫}_{\cos \left(x\right)}^{1\+\sin \left(x\right)}\left(2x^2\+3y^2\+1\right)\ⅆy\ⅆx\=\frac{4}{3}\+\frac{2}{3}{\mathrm{\π}}^3\+2{\mathrm{\π}}^2\+\frac{15}{2}\mathrm{\π}$

Use the MultiInt command from the Student MultivariateCalculus package

$\mathrm{MultiInt}\left(f\,y\=\cos \left(x\right)..1\+\sin \left(x\right)\,x\=0..\mathrm{\π}\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^{\mathrm{\π}}{\∫}_{\cos \left(x\right)}^{1\+\sin \left(x\right)}\left(2x^2\+3y^2\+1\right)\ⅆy\ⅆx$

$\mathrm{MultiInt}\left(f\,y\=\cos \left(x\right)..1\+\sin \left(x\right)\,x\=0..\mathrm{\π}\right)$

$\frac{4}{3}\+\frac{2}{3}{\mathrm{\π}}^3\+2{\mathrm{\π}}^2\+\frac{15}{2}\mathrm{\π}$

Obtain stepwise evaluations via the MultiInt command

$\mathrm{MultiInt}\left(f\,y\=\cos \left(x\right)..1\+\sin \left(x\right)\,x\=0..\mathrm{\π}\,\mathrm{output}\=\mathrm{steps}\right)\:$

${\int }_0^{\mathrm{\pi }}{\int }_{\cos \left(x\right)}^{1\+\sin \left(x\right)}\left(2x^2\+3y^2\+1\right)\ⅆy\ⅆx$

 $\=\frac{4}{3}\+\frac{2}{3}{\mathrm{\pi }}^3\+2{\mathrm{\pi }}^2\+\frac{15}{2}\mathrm{\π}$