Question

Let $f(x,y,z)$be a continuous function of three variables, let localid="1650352548375" ${\Omega }_{yz}=\left\{\right(y,z\left)\right|a\le y\le band{h}_1\left(y\right)\le z\le h_2\left(y\right)\}$be a set of points in the $yz$-plane, and let localid="1650354983053" $\Omega =\left\{\right(x,y,z\left)\right|(y,z)\in {\Omega }_{yz}and{g}_1(y,z)\le x\le g_2(y,z)\}$be a set of points in $3$-space. Find an iterated triple integral equal to the triple integral localid="1650353288865" $\underset{\Omega }{\iiint }f(x,y,z)dV$. How would your answer change iflocalid="1650352747263" ${\Omega }_{yz}=\left\{\right(y,z\left)\right|a\le z\le band{h}_1\left(z\right)\le x\le h_2\left(z\right)\}$?

Short answer

If in $yz$-plane,localid="1650353321170" ${\Omega }_{yz}=\left\{\right(y,z\left)\right|a\le y\le band{h}_1\left(y\right)\le z\le h_2\left(y\right)\}$, then the triple integral becomes,
localid="1650355042503" $\underset{\Omega }{\iiint }f\left(x,y,z\right)dv={\int }_a^b{\int }_{h_1\left(z\right)}^{h_2\left(z\right)}{\int }_{g_1\left(x,y\right)}^{g_2\left(x,y\right)}f\left(x,y,z\right)dxdydz$.

Because, localid="1650355058244" $\Omega =\left\{\right(x,y,z\left)\right|(y,z)\in {\Omega }_{yz}and{g}_1(y,z)\le x\le g_2(y,z)\}$.

Step-by-step solution

Step 1. Given information

$\Omega =\left\{\right(x,y,z\left)\right|(y,z)\in {\Omega }_{yz}and{g}_1(y,z)\le x\le g_2(y,z)\}$.

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

Step 2. Find an iterated integral which is equal to ∭Ωf(x, y,z) dV:

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

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

Because,localid="1650355100648" $\Omega =\left\{\right(x,y,z\left)\right|(y,z)\in {\Omega }_{yz}and{g}_1(y,z)\le x\le g_2(y,z)\}$.