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Find the volume of the region between the cylinder z equals y^2 and the? xy-plane that is bounded by the planes x equals 0 x=0?, x equals 4x=4?, y equals negative 4y=?4?, and y equals 4y=4.

Respuesta :

LammettHash
The volume is given by the triple integral

[tex]\displaystyle\iiint_V\mathrm dV=\int_{x=0}^{x=4}\int_{y=-4}^{y=4}\int_{z=0}^{z=y^2}\mathrm dz\,\mathrm dy\,\mathrm dx[/tex]

which evaluates to

[tex]\displaystyle4\int_{y=-4}^{y=4}y^2\,\mathrm dy=\dfrac{512}3[/tex]

where the coefficient 4 comes from integrating with respect to [tex]x[/tex], and the integrand [tex]y^2[/tex] comes from integrating with respect to [tex]z[/tex].

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