Q. 13 Let 蚁(x,聽y,z)聽be a density fu... [FREE SOLUTION]

Chapter 13: Q. 13 (page 1054)

Let $蚁 (x, y, z)$ be a density function defined on the tetrahedron $惟$ with vertices $(0, 0, 0), (2, 0, 0), (0, 4, 0),$ and $(0, 0, 3)$ . Set up iterated integrals representing the mass of , using all six distinct orders of integration.

Short Answer

Expert verified

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

role="math" localid="1650366865545" ${鈭�}_{0}^{1} {鈭�}_{0}^{3 - \frac{3 x}{2}} {鈭�}_{0}^{4 - 2 x - \frac{4 y}{3}} 蚁 (x, y, z) d y d z d x {鈭�}_{0}^{3} {鈭�}_{0}^{2 - \frac{2 x}{3}} {鈭�}_{0}^{4 - 2 x - \frac{4 y}{3}} 蚁 (x, y, z) d y d x d z {鈭�}_{0}^{4} {鈭�}_{0}^{3 - \frac{3 y}{4}} {鈭�}_{0}^{2 - \frac{y}{2} - \frac{2 z}{3}} 蚁 (x, y, z) d x d z d y {鈭�}_{0}^{3} {鈭�}_{0}^{4 - \frac{4 z}{3}} {鈭�}_{0}^{2 - \frac{y}{2} - \frac{2 z}{3}} 蚁 (x, y, z) d x d y d z {鈭�}_{0}^{4} {鈭�}_{0}^{2 - \frac{y}{2}} {鈭�}_{0}^{3 - \frac{3 x}{2} - \frac{3 y}{4}} 蚁 (x, y, z) d z d x d y$

Step by step solution

Step 1. Given information

Let $蚁 (x, y, z)$ be a density function defined on the tetrahedron $惟$ with vertices localid="1650434213243" $(0, 0, 0), (2, 0, 0), (0, 4, 0),$ and localid="1650434285733" $(0, 0, 3)$ .

Step 2. The other five mass integrals are,

${鈭�}_{0}^{4} {鈭�}_{0}^{2 - \frac{y}{2}} {鈭�}_{0}^{3 - \frac{3 x}{2} - \frac{3 y}{4}} 蚁 (x, y, z) d z d x d y$ .

Since taking $z$ and $x$ after $y$ thelimits from oblique plane becomes,

0 鈮� y 鈮� 4, 0 鈮� x 鈮� 2 - \frac{y}{2} a n d 0 鈮� z 鈮� 1 - \frac{3 x}{2} - \frac{3 y}{4}

Step 3. Similarly the other four mass integrals are,

${鈭�}_{0}^{1} {鈭�}_{0}^{3 - \frac{3 x}{2}} {鈭�}_{0}^{4 - 2 x - \frac{4 y}{3}} 蚁 (x, y, z) d y d z d x {鈭�}_{0}^{3} {鈭�}_{0}^{2 - \frac{2 x}{3}} {鈭�}_{0}^{4 - 2 x - \frac{4 y}{3}} 蚁 (x, y, z) d y d x d z {鈭�}_{0}^{4} {鈭�}_{0}^{3 - \frac{3 y}{4}} {鈭�}_{0}^{2 - \frac{y}{2} - \frac{2 z}{3}} 蚁 (x, y, z) d x d z d y {鈭�}_{0}^{3} {鈭�}_{0}^{4 - \frac{4 z}{3}} {鈭�}_{0}^{2 - \frac{y}{2} - \frac{2 z}{3}} 蚁 (x, y, z) d x d y d z$

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Let $蚁 (x, y, z)$ be a density function defined on the tetrahedron $惟$ with vertices $(0, 0, 0), (2, 0, 0), (0, 4, 0),$ and $(0, 0, 3)$ . Set up iterated integrals representing the mass of , using all six distinct orders of integration.

Short Answer

Step by step solution

Step 1. Given information

Step 2. The other five mass integrals are,

Step 3. Similarly the other four 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),(2,0,0),(0,4,0),and (0,0,3). Set up iterated integrals representing the mass of , using all six distinct orders of integration.

Short Answer

Step by step solution

Step 1. Given information

Step 2. The other five mass integrals are,

Step 3. Similarly the other four mass integrals are,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Probability and Statistics

Logic and Functions

Applied Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Let $蚁 (x, y, z)$ be a density function defined on the tetrahedron $惟$ with vertices $(0, 0, 0), (2, 0, 0), (0, 4, 0),$ and $(0, 0, 3)$ . Set up iterated integrals representing the mass of , using all six distinct orders of integration.