Q. 8 - Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1 . Given information

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

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

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