Q. 14 Let 蚁(x,聽y,z)聽be a density fu... [FREE SOLUTION]

Chapter 13: Q. 14 (page 1054)

Let $蚁 (x, y, z)$ be a density function defined on the tetrahedron $惟$ with vertices $(0, 0, 0), (a, 0, 0), (0, b, 0),$ and $(0, 0, c),$ where $a, b,$ and $c$ are positive real numbers. Set up iterated integrals representing the mass of $惟$ , using all six distinct orders of integration.

Short Answer

Expert verified

The five distinct orders of integration representing the mass of $惟$ is,

role="math" localid="1650435309532" ${鈭�}_{0}^{b} {鈭�}_{0}^{c - \frac{y}{b}} {鈭�}_{0}^{c - \frac{c x}{a} - \frac{c y}{a}} 蚁 (x, y, z) d z d x d y {鈭�}_{0}^{a} {鈭�}_{0}^{b - \frac{c x}{a}} {鈭�}_{0}^{b - \frac{b x}{a} - \frac{b z}{c}} 蚁 (x, y, z) d y d z d x {鈭�}_{0}^{c} {鈭�}_{0}^{b - \frac{b z}{c}} {鈭�}_{0}^{b - \frac{b x}{a} - \frac{b z}{c}} 蚁 (x, y, z) d y d x d z {鈭�}_{0}^{c} {鈭�}_{0}^{b - \frac{b z}{c}} {鈭�}_{0}^{a - \frac{a y}{b} - \frac{a z}{c}} 蚁 (x, y, z) d x d y d z {鈭�}_{0}^{b} {鈭�}_{0}^{c - \frac{c y}{b}} {鈭�}_{0}^{a - \frac{a y}{b} - \frac{a z}{c}} 蚁 (x, y, z) d x d z d y$

Step by step solution

Step 1. Given information

Let $蚁 (x, y, z)$ be a density function defined on the tetrahedron with vertices $(0, 0, 0), (a, 0, 0), (0, b, 0),$ and $(0, 0, c)$ .

Step 2. Similarly the other five mass integrals are,

${鈭�}_{0}^{b} {鈭�}_{0}^{c - \frac{y}{b}} {鈭�}_{0}^{c - \frac{c x}{a} - \frac{c y}{a}} 蚁 (x, y, z) d z d x d y$ .

Since the triangular region becomes $惟_{y z}$ then the ordinate becomes $0 鈮� y 鈮� b, 0 鈮� x 鈮� c - \frac{c}{a} x a n d 0 鈮� z 鈮� c - \frac{c x}{a} - \frac{c y}{a}$ .

The other four mass equations are,

${鈭�}_{0}^{a} {鈭�}_{0}^{b - \frac{c x}{a}} {鈭�}_{0}^{b - \frac{b x}{a} - \frac{b z}{c}} 蚁 (x, y, z) d y d z d x {鈭�}_{0}^{c} {鈭�}_{0}^{b - \frac{b z}{c}} {鈭�}_{0}^{b - \frac{b x}{a} - \frac{b z}{c}} 蚁 (x, y, z) d y d x d z {鈭�}_{0}^{c} {鈭�}_{0}^{b - \frac{b z}{c}} {鈭�}_{0}^{a - \frac{a y}{b} - \frac{a z}{c}} 蚁 (x, y, z) d x d y d z {鈭�}_{0}^{b} {鈭�}_{0}^{c - \frac{c y}{b}} {鈭�}_{0}^{a - \frac{a y}{b} - \frac{a z}{c}} 蚁 (x, y, z) d x d z d y$

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Let $蚁 (x, y, z)$ be a density function defined on the tetrahedron $惟$ with vertices $(0, 0, 0), (a, 0, 0), (0, b, 0),$ and $(0, 0, c),$ where $a, b,$ and $c$ are positive real numbers. Set up iterated integrals representing the mass of $惟$ , using all six distinct orders of integration.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Similarly the other five mass integrals are,

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91影视

Let 蚁(x,y,z)be a density function defined on the tetrahedron 惟with vertices (0,0,0),(a,0,0),(0,b,0),and (0,0,c),where a,b,and care positive real numbers. Set up iterated integrals representing the mass of惟 , using all six distinct orders of integration.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Similarly the other five mass integrals are,

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Discrete Mathematics

Pure Maths

Statistics

Calculus

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Let $蚁 (x, y, z)$ be a density function defined on the tetrahedron $惟$ with vertices $(0, 0, 0), (a, 0, 0), (0, b, 0),$ and $(0, 0, c),$ where $a, b,$ and $c$ are positive real numbers. Set up iterated integrals representing the mass of $惟$ , using all six distinct orders of integration.