Chapter 13: Q. 40 (page 1055)

Describe the three-dimensional region expressed in each iterated integral in Exercises 35–44.
$\int_{0}^{2} \int_{0}^{1 - 3 x / 2} \int_{2 x + (4 / 3) y - 4}^{0} f (x, y, z) d z d y d x$

Short Answer

Expert verified

The three-dimensional region is given by planer equation,

$\frac{x}{2} + \frac{y}{3} - \frac{z}{4} = 1$

Step by step solution

Step 1. Given Information

We are given,

$\int_{0}^{2} \int_{0}^{1 - 3 x / 2} \int_{2 x + (4 / 3) y - 4}^{0} f (x, y, z) d z d y d x$

Step 2. The three dimensional region.

By the definition of triple integral $\int_{a_{1}}^{a_{1}} \int_{b_{4}}^{b_{2}} \int_{a_{1}}^{c_{2}} f (x, y, z) d z d y d x$ 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

The given integral $\int_{0}^{2} \int_{0}^{1 - \frac{3 x}{2}} \int_{2 x + \frac{4}{3} y - 4}^{0} f (x, y, z) d z d y d x$ represents the volume of the tetrahedron whose vertices are $(0, 0, 0) (2, 0, 0) (0, 3, 0) (0, 0, - 4)$

Since from the given integral

$z = 2 x + \frac{4}{3} y - 4 \Rightarrow 6 x + 4 y - 3 z = 12 \frac{x}{2} + \frac{y}{3} - \frac{z}{4} = 1$

From this planer equation the vertices of the tetrahedron becomes $(0, 0, 0) (2, 0, 0) (0, 3, 0) (0, 0, - 4)$ .

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Describe the three-dimensional region expressed in each iterated integral in Exercises 35–44.
$\int_{0}^{2} \int_{0}^{1 - 3 x / 2} \int_{2 x + (4 / 3) y - 4}^{0} f (x, y, z) d z d y d x$

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.∫02∫01-3x/2∫2x+(4/3)y-40f(x,y,z)dzdydx

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

Calculus

Theoretical and Mathematical Physics

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Applied Mathematics

Study anywhere. Anytime. Across all devices.

Describe the three-dimensional region expressed in each iterated integral in Exercises 35–44.
$\int_{0}^{2} \int_{0}^{1 - 3 x / 2} \int_{2 x + (4 / 3) y - 4}^{0} f (x, y, z) d z d y d x$