Question

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}

Answer 1

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`