Q. 10 - Calculus

Question

Let $f(x, y,z)$ be a continuous function of three variables, let ${\Omega }_{xz}=\left\{\right(x,z\left)\right|a\le x\le b and h_1\left(x\right)\le z\le h_2\left(x\right)\}$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 ${\Omega }_{xz}=\left\{\right(x,z\left)\right|a\le z\le b and h_1\left(z\right)\le x\le h_2\left(z\right)\}$?

Step-by-Step Solution

Answer

If in $xz$-plane ${\Omega }_{xz}=\left\{\right(x,z\left)\right|a\le x\le b and h_1\left(x\right)\le z\le h_2\left(x\right)\}$,then the triple integral becomes,

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

Step 1 . Given information

${\Omega }_{xz}=\left\{\right(x,z\left)\right|a\le x\le b and h_1\left(x\right)\le z\le h_2\left(x\right)\}$.

Step 2 . Find an iterated triple integral which is equal to ∭ Ω   f x , y , z d V :

If in $xz$-plane ${\Omega }_{xz}=\left\{\right(x,z\left)\right|a\le x\le b and h_1\left(x\right)\le z\le h_2\left(x\right)\}$, then the triple integral becomes,

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