Let ρ(x, y,z) be a density function defined on the rectangular solid R where role="math" localid="1650360487967" R={(x,y,z)|a1≤x≤a2,b1≤y≤b2, and c1≤z≤c2}. Set up iterated integrals representing the mass of R, using all six distinct orders of integration.

The six distinct orders of integration representing the mass of R are,role="math" localid="1650361542302" ∫a1a2∫b1b2∫c1c2 ρx,y,zdzdydx∫b1b2 ∫a1a2∫c1c2ρx,y,zdzdxdy ∫a1a2∫c1c2∫b1b2 ρx,y,zdydzdx ...

Q. 12 - Calculus | StudyQuestionHub

Q. 12

Question

Let $ρ (x, y, z)$ be a density function defined on the rectangular solid $R$ where $R = {(x, y, z) | a_{1} \leq x \leq a_{2}, b_{1} \leq y \leq b_{2}, a n d c_{1} \leq z \leq c_{2}}$ . Set up iterated integrals representing the mass of $R$ , using all six distinct orders of integration.

Step-by-Step Solution

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Answer

The six distinct orders of integration representing the mass of $R$ are,

$\int_{a_{1}}^{a_{2}} \int_{b_{1}}^{b_{2}} \int_{c_{1}}^{c_{2}} ρ (x, y, z) d z d y d x \int_{b_{1}}^{b_{2}} \int_{a_{1}}^{a_{2}} \int_{c_{1}}^{c_{2}} ρ (x, y, z) d z d x d y \int_{a_{1}}^{a_{2}} \int_{c_{1}}^{c_{2}} \int_{b_{1}}^{b_{2}} ρ (x, y, z) d y d z d x \int_{c_{1}}^{c_{2}} \int_{a_{1}}^{a_{2}} \int_{b_{1}}^{b_{2}} ρ (x, y, z) d y d x d z \int_{b_{1}}^{b_{2}} \int_{c_{1}}^{c_{2}} \int_{a_{1}}^{a_{2}} ρ (x, y, z) d x d z d y \int_{c_{1}}^{c_{2}} \int_{b_{1}}^{b_{2}} \int_{a_{1}}^{a_{2}} ρ (x, y, z) d x d y d z$

1Step 1 . Given information

$R = {(x, y, z) | a_{1} \leq x \leq a_{2}, b_{1} \leq y \leq b_{2}, a n d c_{1} \leq z \leq c_{2}}$ .

2Step 2 . The definition of the iterated triple integral:

If $ρ (x, y, z)$ be a density function defined on the rectangular solid $R$ where $R = {(x, y, z) | a \leq x \leq b, c \leq y \leq d, a n d e \leq z \leq f}$ then the mass of $R$ is defined as, $\underset{Ω}{∭} ρ (x, y, z) d V = \int_{a}^{b} \int_{c}^{d} \int_{e}^{f} ρ (x, y, z) d z d y d x$ .

The other mass equations are defined in the same way for other $5$ triple integrals. Given that $ρ (x, y, z)$ be a density function defined on the rectangular solid $R$ where $R = {(x, y, z) | a_{1} \leq x \leq a_{2}, b_{1} \leq y \leq b_{2}, a n d c_{1} \leq z \leq c_{2}}$ then the mass of $R$ is defined as $\underset{Ω}{∭} ρ (x, y, z) d V = \int_{a_{1}}^{a_{2}} \int_{b_{1}}^{b_{2}} \int_{c_{1}}^{c_{2}} ρ (x, y, z) d z d y d x$ .

3Step 3 . The other five equations for the mass are,

$\int_{b_{1}}^{b_{2}} \int_{a_{1}}^{a_{2}} \int_{c_{1}}^{c_{2}} ρ (x, y, z) d z d x d y \int_{a_{1}}^{a_{2}} \int_{c_{1}}^{c_{2}} \int_{b_{1}}^{b_{2}} ρ (x, y, z) d y d z d x \int_{c_{1}}^{c_{2}} \int_{a_{1}}^{a_{2}} \int_{b_{1}}^{b_{2}} ρ (x, y, z) d y d x d z \int_{b_{1}}^{b_{2}} \int_{c_{1}}^{c_{2}} \int_{a_{1}}^{a_{2}} ρ (x, y, z) d x d z d y \int_{c_{1}}^{c_{2}} \int_{b_{1}}^{b_{2}} \int_{a_{1}}^{a_{2}} ρ (x, y, z) d x d y d z$

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