Chapter 13: Q. 11 (page 1054)

Let $ρ (x, y, z)$ be a density function defined on the rectangular solid $R$ where role="math" localid="1650358913892" $R = {(x, y, z) | - 1 \leq x \leq 3, 0 \leq y \leq 2, a n d 2 \leq z \leq 7}$ . Set up iterated integrals representing the mass of $R$ , using all six distinct orders of integration.

Short Answer

Expert verified

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

role="math" localid="1650359097828" $\int_{- 1}^{3} \int_{0}^{2} \int_{2}^{7} ρ (x, y, z) d z d y d x \int_{0}^{2} \int_{- 1}^{3} \int_{2}^{7} ρ (x, y, z) d z d x d y \int_{- 1}^{3} \int_{2}^{7} \int_{0}^{2} ρ (x, y, z) d y d z d x \int_{2}^{7} \int_{- 1}^{3} \int_{0}^{2} ρ (x, y, z) d y d x d z \int_{0}^{2} \int_{2}^{7} \int_{- 1}^{3} ρ (x, y, z) d x d z d y \int_{2}^{7} \int_{0}^{2} \int_{- 1}^{3} ρ (x, y, z) d x d y d z$

Step by step solution

Step 1. Given information

$R = {(x, y, z) | - 1 \leq x \leq 3, 0 \leq y \leq 2, a n d 2 \leq z \leq 7}$ .

Step 2. The definition of the iterated triple integral:

If $ρ (x, y, z)$ be a density function defined on the rectangular solid $R$ where localid="1650361207932" $R = {(x, y, z) | a \leq x \leq b, c \leq y \leq d, a n d e \leq z \leq f}$ , then the mass of $R$ is defined as, localid="1650361158631" $\underset{Ω}{∭} ρ (x, y, z) d V = \int_{a}^{b} \int_{c}^{d} \int_{e}^{f} ρ (x, y, z) d z d y d x$ .

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 $R = {(x, y, z) | - 1 \leq x \leq 3, 0 \leq y \leq 2, a n d 2 \leq z \leq 7}$ , then the mass of the $R$ is defined as, $\underset{Ω}{∭} ρ (x, y, z) d V = \int_{- 1}^{3} \int_{0}^{2} \int_{2}^{7} ρ (x, y, z) d z d y d x$ .

Step 3. The other five distinct order of integration is,

$\int_{0}^{2} \int_{- 1}^{3} \int_{2}^{7} ρ (x, y, z) d z d x d y \int_{- 1}^{3} \int_{2}^{7} \int_{0}^{2} ρ (x, y, z) d y d z d x \int_{2}^{7} \int_{- 1}^{3} \int_{0}^{2} ρ (x, y, z) d y d x d z \int_{0}^{2} \int_{2}^{7} \int_{- 1}^{3} ρ (x, y, z) d x d z d y \int_{2}^{7} \int_{0}^{2} \int_{- 1}^{3} ρ (x, y, z) d x d y d z$

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Short Answer

Step by step solution

Step 1. Given information

Step 2. The definition of the iterated triple integral:

Step 3. The other five distinct order of integration is,

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Let ρ(x,y,z)be a density function defined on the rectangular solid Rwhere role="math" localid="1650358913892" R={(x,y,z)|−1≤x≤3,0≤y≤2,and2≤z≤7}. Set up iterated integrals representing the mass of R, using all six distinct orders of integration.

Short Answer

Step by step solution

Step 1. Given information

Step 2. The definition of the iterated triple integral:

Step 3. The other five distinct order of integration is,

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Geometry

Statistics

Theoretical and Mathematical Physics

Calculus

Discrete Mathematics

Study anywhere. Anytime. Across all devices.