Chapter 13: Q. 37 (page 1055)

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

Short Answer

Expert verified

The three-dimensional region is,

$ℝ = {(x, y, z) / 0 \leq x \leq 3, 0 \leq y \leq 3, 0 \leq z \leq 3 - y}$

Step by step solution

Step 1. Given Information

We are given,

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

Step 2. The three dimensional region.

By the definition of triple integral $\int_{a_{1}}^{a_{1}} \int_{b_{1}}^{b_{2}} \int_{c_{1}}^{c_{2}} f (x, y, z) d z d y d x$ 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 iterated triple integral $\int_{0}^{3} \int_{0}^{3} \int_{0}^{3 - y} f (x, y, z) d z d y d x$ represents the volume of the half cube represents by $ℝ = {(x, y, z) / 0 \leq x \leq 3, 0 \leq y \leq 3, 0 \leq z \leq 3 - y}$

Or the planer equation $x + y + z = 3$ .

Since from the given triple integral the limits are $0 \leq x \leq 3, 0 \leq y \leq 3, 0 \leq z \leq 3 - y$ .

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

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.∫03∫03∫03-yf(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

Theoretical and Mathematical Physics

Logic and Functions

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Pure Maths

Study anywhere. Anytime. Across all devices.

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