Question

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

Short answer

If ${\Omega }_{xy}=\left\{\right(x,y\left)\right|a\le y\le band{h}_1\left(y\right)\le x\le h_2\left(y\right)\}$then the triple integral becomes,role="math" localid="1650351304797" $\underset{\Omega }{\iiint f}\left(x,y,z\right)dv={\int }_a^b{\int }_{h_1\left(y\right)}^{h_2\left(y\right)}{\int }_{g_1\left(x,y\right)}^{g_2\left(x,y\right)}dzdxdy$.

Step-by-step solution

Step 1. Given information

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

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

If ${\Omega }_{xy}=\left\{\right(x,y\left)\right|a\le y\le band{h}_1\left(y\right)\le x\le h_2\left(y\right)\}$, then the triple integral becomes $\underset{\Omega }{\iiint f}\left(x,y,z\right)dv={\int }_a^b{\int }_{h_1\left(y\right)}^{h_2\left(y\right)}{\int }_{g_1\left(x,y\right)}^{g_2\left(x,y\right)}dzdxdy$.

Since$a\le y\le b$and$h_1\left(y\right)\le x\le h_2\left(y\right)$and$g_1\left(x,y\right)\le z\le g_2\left(x,y\right)$.