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Calculate the volume of the solid, bounded by the surfaces: z=4x2 +4y2; z=x2+y2; z=4.

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In cylindrical coordinates, the volume is given by the integral


\displaystyle\int_(\theta=0)^(\theta=2\pi)\int_(z=0)^(z=4)\int_(r=√(z/4))^(r=\sqrt z)r\,\mathrm dr\,\mathrm dz\,\mathrm d\theta

=\displaystyle2\pi\int_(z=0)^(z=4)\int_(r=√(z/4))^(r=\sqrt z)r\,\mathrm dr\,\mathrm dz

image

=\displaystyle\pi\int_(z=0)^(z=4)\left(z-\frac z4\right)\,\mathrm dz

=\displaystyle\frac{3\pi}4\int_(z=0)^(z=4)z\,\mathrm dz

=\displaystyle\frac{3\pi}8z^2\bigg|_(z=0)^(z=4)

=6\pi
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