Chapter 13: Q. 39 (page 1055)

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

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,

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

Step 2. The three dimensional region.

By the definition of triple integral $\int_{a_{1}}^{a_{1}} \int_{b_{1}}^{b_{2}} \int_{c_{1}}^{c_{2}} f (x, y, z) d z d y d x$ represent the volume of the solid region $ℝ = \{(x, y, z) ∣ a_{1} \leq x \leq a_{2}, b_{1} \leq y \leq b_{2}, c_{1} \leq z \leq c_{2}\}$

Using this definition, we get

Given triple integral $\int_{0}^{3} \int_{0}^{1 - \frac{y}{3}} \int_{0}^{2 - \frac{2}{3} y - 2 x} f (x, y, z) d x d z d y$ 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 $0 \leq y \leq 3, 0 \leq x \leq 1 - \frac{y}{3}, 0 \leq z \leq 2 - \frac{2}{3} y - 2 x$

The planer equation becomes

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

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

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 35–44.∫03∫01-y/3∫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.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Statistics

Geometry

Mechanics Maths

Theoretical and Mathematical Physics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

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