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Loading Student:-MultivariateCalculus

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

$\mathrm{YB}\=x^2-6$$\stackrel{\text{assign}}{\to }$

$\mathrm{YT}\=-x$$\stackrel{\text{assign}}{\to }$

Access the MultiInt command via the Context Panel

$f$ = $2x^2\+3y^2\+1$$\stackrel{\text{MultiInt}}{\to }$${\∫}_{-3}^2{\∫}_{x^2-6}^{-x}\left(2x^2\+3y^2\+1\right)\ⅆy\ⅆx$$\=$$\frac{48625}{84}$$$

Tools≻Tasks≻Browse:

Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻

Table 5.3.6(a)   Visualizing $R$ and the resulting volume for iteration in the order $\mathrm{dy} \mathrm{dx}$ 

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

${\int }_{-3}^2{\int }_{x^2-6}^{-x}f \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $\frac{48625}{84}$$$

Display the iterated integrals

${\int }_{-3}^2{\int }_{x^2-6}^{-x}f \mathit{\ⅆ}y \mathit{\ⅆ}x$

${\∫}_{-3}^2{\∫}_{x^2-6}^{-x}\left(2x^2\+3y^2\+1\right)\ⅆy\ⅆx$

$\=$

$\frac{48625}{84}$

Initialize

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

$f≔1\+2 x^2\+3 y^2\:$

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

$\mathrm{Int}\left(f\,\left[y\=x^2-6..-x\,x\=-3..2\right]\right)\=\mathrm{int}\left(f\,\left[y\=x^2-6..-x\,x\=-3..2\right]\right)$

${\∫}_{-3}^2{\∫}_{x^2-6}^{-x}\left(2x^2\+3y^2\+1\right)\ⅆy\ⅆx\=\frac{48625}{84}$

Use the MultiInt command from the Student MultivariateCalculus package

$\mathrm{MultiInt}\left(f\,y\=x^2-6..-x\,x\=-3..2\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_{-3}^2{\∫}_{x^2-6}^{-x}\left(2x^2\+3y^2\+1\right)\ⅆy\ⅆx$

$\mathrm{MultiInt}\left(f\,y\=x^2-6..-x\,x\=-3..2\right)$

$\frac{48625}{84}$

Obtain stepwise evaluations via the MultiInt command

$\mathrm{MultiInt}\left(f\,y\=x^2-6..-x\,x\=-3..2\,\mathrm{output}\=\mathrm{steps}\right)$

$\begin{array}{ccc}\multicolumn{3}{c}{{\int }_{\mathrm{-3}}^2{\int }_{x^2-6}^{-x}\left(2x^2+3y^2+1\right)\ⅆy\ⅆx}\\ & \text{=}& {\int }_{\mathrm{-3}}^2\left(\genfrac{}{}{0ex}{}{\left(2x^{2}y+y^3+y\right)}{\phantom{y=x^2-6..-x}}|\genfrac{}{}{0ex}{}{\phantom{\left(2x^{2}y+y^3+y\right)}}{y=x^2-6..-x}\right)\ⅆx\hfill \\ & \text{=}& {\int }_{\mathrm{-3}}^2\left(2x^2\left(-x^2-x+6\right)-x^3-{\left(x^2-6\right)}^3-x^2-x+6\right)\ⅆx\hfill \\ & \text{=}& \genfrac{}{}{0ex}{}{\left(\frac{16}{5}{x}^5-\frac{3}{4}{x}^4-\frac{97}{3}{x}^3-\frac{1}{7}{x}^7+222x-\frac{1}{2}{x}^2\right)}{\phantom{x=\mathrm{-3}..2}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{16}{5}{x}^5-\frac{3}{4}{x}^4-\frac{97}{3}{x}^3-\frac{1}{7}{x}^7+222x-\frac{1}{2}{x}^2\right)}}{x=\mathrm{-3}..2}\hfill \end{array}$