3. let $u$ be the solid enclosed by the paraboloids $z = x^{2}+y^{2}$ and $z = 8-(x^{2}+y^{2})$. (note: the…

3. let $u$ be the solid enclosed by the paraboloids $z = x^{2}+y^{2}$ and $z = 8-(x^{2}+y^{2})$. (note: the paraboloids intersect where $z = 4$.) write $iiint_{u}f(x,y,z) dv$ as an iterated integral in the order $dz dy dx$.

3. let $u$ be the solid enclosed by the paraboloids $z = x^{2}+y^{2}$ and $z = 8-(x^{2}+y^{2})$. (note: the paraboloids intersect where $z = 4$.) write $iiint_{u}f(x,y,z) dv$ as an iterated integral in the order $dz dy dx$.

Answer

Explanation:

Step1: Find the range of (z)

The lower - bound of (z) is given by (z = x^{2}+y^{2}) and the upper - bound is (z = 8-(x^{2}+y^{2})).

Step2: Find the intersection of the paraboloids in the (xy) - plane

Set (x^{2}+y^{2}=8-(x^{2}+y^{2})), then (2(x^{2}+y^{2}) = 8), or (x^{2}+y^{2}=4). To find the range of (y) in terms of (x), we solve (y^{2}=4 - x^{2}), so (y=-\sqrt{4 - x^{2}}) to (y = \sqrt{4 - x^{2}}).

Step3: Find the range of (x)

The circle (x^{2}+y^{2}=4) in the (xy) - plane has (x) ranging from (x=-2) to (x = 2).

Step4: Write the triple - integral

The triple - integral (\iiint_{U}f(x,y,z)dV) in the order (dzdydx) is (\int_{-2}^{2}\int_{-\sqrt{4 - x^{2}}}^{\sqrt{4 - x^{2}}}\int_{x^{2}+y^{2}}^{8-(x^{2}+y^{2})}f(x,y,z)dzdydx).

Answer:

(\int_{-2}^{2}\int_{-\sqrt{4 - x^{2}}}^{\sqrt{4 - x^{2}}}\int_{x^{2}+y^{2}}^{8-(x^{2}+y^{2})}f(x,y,z)dzdydx)