Q. 39 Describe the three-dimensional r... [FREE SOLUTION]

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Chapter 13: Q. 39 (page 1055)

Describe the three-dimensional region expressed in each iterated integral in Exercises 35鈥�44.
${鈭�}_{0}^{3} {鈭�}_{0}^{1 - y / 3} {鈭�}_{0}^{2 - (2 / 3) y - 2 z} f (x, y, z) d x d z d y$

Short Answer

Expert verified

The three-dimensional region is given by planer equation,

$\frac{x}{1} + \frac{y}{3} + \frac{z}{2} = 1$

Step by step solution

Step 1. Given Information

We are given,

${鈭�}_{0}^{3} {鈭�}_{0}^{1 - y / 3} {鈭�}_{0}^{2 - (2 / 3) y - 2 z} f (x, y, z) d x d z d y$

Step 2. The three dimensional region.

By the definition of triple integral ${鈭�}_{a_{1}}^{a_{1}} {鈭�}_{b_{1}}^{b_{2}} {鈭�}_{c_{1}}^{c_{2}} f (x, y, z) d z d y d x$ represent the volume of the solid region $鈩� = \{(x, y, z) 鈭� a_{1} 鈮� x 鈮� a_{2}, b_{1} 鈮� y 鈮� b_{2}, c_{1} 鈮� z 鈮� c_{2}\}$

Using this definition, we get

Given triple integral ${鈭�}_{0}^{3} {鈭�}_{0}^{1 - \frac{y}{3}} {鈭�}_{0}^{2 - \frac{2}{3} y - 2 x} f (x, y, z) d x d z d y$ represents the volume of the tetrahedron whose vertices are $(0, 0, 0) (2, 0, 0) (0, 3, 0) (0, 0, 1)$ .

Since from the given integral the limits are $0 鈮� y 鈮� 3, 0 鈮� x 鈮� 1 - \frac{y}{3}, 0 鈮� z 鈮� 2 - \frac{2}{3} y - 2 x$

The planer equation becomes

$z = 2 - \frac{2}{3} y - 2 x 鈬� 2 x + \frac{2}{3} y + z = 2 \frac{x}{1} + \frac{y}{3} + \frac{z}{2} = 1$

It represents the region of the tetrahedron with vertices $(0, 0, 0) (2, 0, 0) (0, 3, 0) (0, 0, 1)$ .

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91影视

Describe the three-dimensional region expressed in each iterated integral in Exercises 35鈥�44.
${鈭�}_{0}^{3} {鈭�}_{0}^{1 - y / 3} {鈭�}_{0}^{2 - (2 / 3) y - 2 z} f (x, y, z) d x d z d y$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. The three dimensional region.

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Describe the three-dimensional region expressed in each iterated integral in Exercises 35鈥�44.鈭�03鈭�01-y/3鈭�02-(2/3)y-2zf(x,y,z)dxdzdy

Short Answer

Step by step solution

Step 1. Given Information

Step 2. The three dimensional region.

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Applied Mathematics

Theoretical and Mathematical Physics

Discrete Mathematics

Decision Maths

Geometry

Study anywhere. Anytime. Across all devices.

Describe the three-dimensional region expressed in each iterated integral in Exercises 35鈥�44.
${鈭�}_{0}^{3} {鈭�}_{0}^{1 - y / 3} {鈭�}_{0}^{2 - (2 / 3) y - 2 z} f (x, y, z) d x d z d y$