Question

Let $f(x,y,z)$be a continuous function of three variables, let role="math" localid="1650355225242" ${\Omega }_{xz}=\left\{\right(x,z\left)\right|a\le x\le band{h}_1\left(x\right)\le z\le h_2\left(x\right)\}$be a set of points in the $xz$-plane, and let $\Omega =\left\{\right(x,y,z\left)\right|(x,z)\in {\Omega }_{xz}and{g}_1(x,z)\le y\le g_2(x,z)\}$be a set of points in 3-space. Find an iterated triple integral equal to the triple integral $\underset{\Omega }{\iiint }f\left(x,y,z\right)dV$. How would your answer change if${\Omega }_{xz}=\left\{\right(x,z\left)\right|a\le z\le band{h}_1\left(z\right)\le x\le h_2\left(z\right)\}$?

Short answer

If in $xz$-plane ${\Omega }_{xz}=\left\{\right(x,z\left)\right|a\le x\le band{h}_1\left(x\right)\le z\le h_2\left(x\right)\}$,then the triple integral becomes,

role="math" localid="1650355536570" $\underset{\Omega }{\iiint }f\left(x,y,z\right)dv={\int }_a^b{\int }_{h_1\left(z\right)}^{h_2\left(z\right)}{\int }_{g_1\left(x,z\right)}^{g_2\left(x,z\right)}f\left(x,y,z\right)dydxdz$.

Step-by-step solution

Step 1. Given information

${\Omega }_{xz}=\left\{\right(x,z\left)\right|a\le x\le band{h}_1\left(x\right)\le z\le h_2\left(x\right)\}$.

Step 2. Find an iterated triple integral which is equal to ∭Ω fx,y,zdV:

If in $xz$-plane ${\Omega }_{xz}=\left\{\right(x,z\left)\right|a\le x\le band{h}_1\left(x\right)\le z\le h_2\left(x\right)\}$, then the triple integral becomes,

$\underset{\Omega }{\iiint }f\left(x,y,z\right)dv={\int }_a^b{\int }_{h_1\left(z\right)}^{h_2\left(z\right)}{\int }_{g_1\left(x,z\right)}^{g_2\left(x,z\right)}f\left(x,y,z\right)dydxdz$.