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Loading Student:-MultivariateCalculus

$z_1\=7-x^2-y^2$$\stackrel{\text{assign}}{\to }$$$

  $z_2\=2 x\+4 y-6$$\stackrel{\text{assign}}{\to }$$$

Access the MultiInt command via the Context Panel

$z_1-z_2$ = $-x^2-y^2-2x-4y\+13$$\stackrel{\text{MultiInt}}{\to }$${\∫}_0^1{\∫}_x^{-x^2\+2}\left(-x^2-y^2-2x-4y\+13\right)\ⅆy\ⅆx$$\=$$\frac{782}{105}$

Tools≻Tasks≻Browse:

Calculus - Vector≻Integration≻Multiple Integration≻2-D≻Over a General 2-D Region

Table 6.2.10(a)   Solution by task template

Write an appropriate iterated integral and evaluate

${\int }_0^1{\int }_x^{2-x^2}\left(z_1-z_2\right) \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $\frac{782}{105}$$$$$

Table 6.2.10(b)   Solution from first principles

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

$$\begin{gathered} z_1≔7-x^2-y^2\: \\ {z}_2≔2 x\+4 y-6\: \end{gathered}$$

Top-level, using the Int and int commands

$\mathrm{Int}\left(z_1-z_2\,\left[y\=x..2-x^2\,x\=0..1\right]\right)\=\mathrm{int}\left(z_1-z_2\,\left[y\=x..2-x^2\,x\=0..1\right]\right)\mathit{ }$

${\∫}_0^1{\∫}_x^{-x^2\+2}\left(-x^2-y^2-2x-4y\+13\right)\ⅆy\ⅆx\=\frac{782}{105}$

Use the MultiInt command from the Student MultivariateCalculus package

$\mathrm{MultiInt}\left(z_1-z_2\,y\=x..2-x^2\,x\=0..1\right)$ = $\frac{782}{105}$$$

$\mathrm{MultiInt}\left(z_1-z_2\,y\=x..2-x^2\,x\=0..1\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^1{\∫}_x^{-x^2\+2}\left(-x^2-y^2-2x-4y\+13\right)\ⅆy\ⅆx$

Use the MultiInt command with a pre-defined domain option

$\mathrm{MultiInt}\left(z_1-z_2\,\left[x\,y\right]\=\mathrm{Region}\left(0..1\,x..2-x^2\right)\right)$ = $\frac{782}{105}$$$

$\mathrm{MultiInt}\left(z_1-z_2\,\left[x\,y\right]\=\mathrm{Region}\left(0..1\,x..2-x^2\right)\,\mathrm{output}\=\mathrm{integral}\right)$

${\∫}_0^1{\∫}_x^{-x^2\+2}\left(-x^2-y^2-2x-4y\+13\right)\ⅆy\ⅆx$