Chapter 13: Q. 43 (page 1055)

Describe the three-dimensional region expressed in each iterated integral in Exercises 35–44.
$\int_{- 3}^{3} \int_{- \sqrt{9 - y^{2}}}^{\sqrt{9 - y^{2}}} \int_{0}^{9 - y^{2} - z^{2}} f (x, y, z) d x d z d y$

Short Answer

Expert verified

The three-dimensional region represents the volume of the region is the right by the parabolic $x = 9 - y^{2} - z^{2}$ bounded on the left by the yz plane. ,

Step by step solution

Step 1. Given Information.

We are given,

$\int_{- 3}^{3} \int_{- \sqrt{9 - y^{2}}}^{\sqrt{9 - y^{2}}} \int_{0}^{9 - y^{2} - z^{2}} f (x, y, z) d x d z d y$

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

The given triple integral represents the volume of the region is the right by the parabolic $x = 9 - y^{2} - z^{2}$ bounded on the left by the yz plane.

Since from the given integral we observe that

$x = 9 - y^{2} - z^{2}$

Which represent the parabolic region and

$z = \sqrt{9 - y^{2}}$

It represent yz plane.

Thus, the given iterated integral represents the volume of the region is the right by the parabolic $x = 9 - y^{2} - z^{2}$ bounded on the left by the yz plane.

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Describe the three-dimensional region expressed in each iterated integral in Exercises 35–44.
$\int_{- 3}^{3} \int_{- \sqrt{9 - y^{2}}}^{\sqrt{9 - y^{2}}} \int_{0}^{9 - y^{2} - z^{2}} f (x, y, z) d x d z d y$

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.∫-33∫-9-y29-y2∫09-y2-z2f(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

Mechanics Maths

Probability and Statistics

Calculus

Decision Maths

Discrete Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Describe the three-dimensional region expressed in each iterated integral in Exercises 35–44.
$\int_{- 3}^{3} \int_{- \sqrt{9 - y^{2}}}^{\sqrt{9 - y^{2}}} \int_{0}^{9 - y^{2} - z^{2}} f (x, y, z) d x d z d y$