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Express the triple integral f(x,y,z dv as an iterated integral in the two orders dz dy dx and dz dx dy where e = { (x,y,z sqrt(x^2 y^2 < z <1}

User Almudhafar
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1 Answer

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Assuming
E=\{(x,y,z)~:~√(x^2+y^2)<z<1\}, so that the region is essentially a cone with base radius 1 and height 1, the triple integral of
f(x,y,z) can be expressed as


\displaystyle\iiint_Ef(x,y,z)\,\mathrm dV=\int_(x=-1)^(x=1)\int_(y=-√(1-x^2))^(y=√(1-x^2))\int_(z=√(x^2+y^2))^(z=1)f(x,y,z)\,\mathrm dz\,\mathrm dy\,\mathrm dx

or


\displaystyle\iiint_Ef(x,y,z)\,\mathrm dV=\int_(y=-1)^(y=1)\int_(x=-√(1-y^2))^(x=√(1-y^2))\int_(z=√(x^2+y^2))^(z=1)f(x,y,z)\,\mathrm dz\,\mathrm dx\,\mathrm dy
User Chris Poe
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7.9k points
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