Chapter 13: Q. 36 (page 1055)

Describe the three-dimensional region expressed in each iterated integral in Exercises 35–44.
$\int_{- 1}^{5} \int_{- 3}^{2} \int_{4}^{8} f (x, y, z) d z d x d y$

Short Answer

Expert verified

The three-dimensional region is,

$ℝ = {(x, y, z) ∣ - 3 \leq x \leq 2, - 1 \leq y \leq 5, 4 \leq z \leq 8}$

Step by step solution

Step 1. Given Information

We are given,

$\int_{- 1}^{5} \int_{- 3}^{2} \int_{4}^{8} f (x, y, z) d z d x d y$

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,

The given iterated triple integral $\int_{- 1}^{5} \int_{- 3}^{2} \int_{4}^{8} f (x, y, z) d z d x d y$ represents the volume of the rectangular solid given by $ℝ = {(x, y, z) ∣ - 3 \leq x \leq 2, - 1 \leq y \leq 5, 4 \leq z \leq 8}$

Since by taking the order of the integration in the given iterated integral order that is $d z d x d y$ .

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Describe the three-dimensional region expressed in each iterated integral in Exercises 35–44.
$\int_{- 1}^{5} \int_{- 3}^{2} \int_{4}^{8} f (x, y, z) d z d x d y$

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.∫-15∫-32∫48f(x,y,z)dzdxdy

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

Discrete Mathematics

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Mechanics Maths

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Describe the three-dimensional region expressed in each iterated integral in Exercises 35–44.
$\int_{- 1}^{5} \int_{- 3}^{2} \int_{4}^{8} f (x, y, z) d z d x d y$