Q. 9 - Calculus

Question

Let $f(x, y,z)$ be a continuous function of three variables, let ${\Omega }_{yz}=\left\{\right(y,z\left)\right|a\le y\le b and h_1\left(y\right)\le z\le h_2\left(y\right)\}$ be a set of points in the $yz$-plane, and let $\Omega =\left\{\right(x,y,z\left)\right|(y,z)\in {\Omega }_{yz} and g_1(y,z)\le x\le g_2(y,z)\}$ be a set of points in $3$-space. Find an iterated triple integral equal to the triple integral $\underset{\Omega }{\iiint }f(x, y,z) dV$. How would your answer change if ${\Omega }_{yz}=\left\{\right(y,z\left)\right|a\le z\le b and h_1\left(z\right)\le x\le h_2\left(z\right)\}$?

Step-by-Step Solution

Answer

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

Because, $\Omega =\left\{\right(x,y,z\left)\right|(y,z)\in {\Omega }_{yz} and g_1(y,z)\le x\le g_2(y,z)\}$.

Step 1 . Given information

$\Omega =\left\{\right(x,y,z\left)\right|(y,z)\in {\Omega }_{yz} and g_1(y,z)\le x\le g_2(y,z)\}$.

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

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

If in $yz$-plane ${\Omega }_{yz}=\left\{\right(y,z\left)\right|a\le y\le b and h_1\left(y\right)\le z\le h_2\left(y\right)\}$, 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, $\Omega =\left\{\right(x,y,z\left)\right|(y,z)\in {\Omega }_{yz} and g_1(y,z)\le x\le g_2(y,z)\}$.