Q. 44 Describe the three-dimensional r... [FREE SOLUTION]

Chapter 13: Q. 44 (page 1055)

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

Short Answer

Expert verified

The three-dimensional region represents the volume of the region is the right by the parabolic $(y - 3)^{2} = x^{2} + z^{2}$ bounded on the left by the xz plane. ,

Step by step solution

Step 1. Given Information.

We are given,

${鈭�}_{- 3}^{3} {鈭�}_{- \sqrt{9 - z^{2}}}^{\sqrt{9 - z^{2}}} {鈭�}_{0}^{3 - \sqrt{x^{2} + z^{2}}} f (x, y, z) d y d x d z$

Step 2. The three dimensional region.

By the definition of triple integral ${鈭�}_{a_{1}}^{a_{1}} {鈭�}_{b_{1}}^{b_{2}} {鈭�}_{c_{1}}^{c_{2}} f (x, y, z) d z d y d x$ represent the volume of the solid region $鈩� = \{(x, y, z) 鈭� a_{1} 鈮� x 鈮� a_{2}, b_{1} 鈮� y 鈮� b_{2}, c_{1} 鈮� z 鈮� c_{2}\}$

Using this definition, we get

The given triple integral ${鈭�}_{- 3}^{3} {鈭�}_{- \sqrt{9 - y^{2}}}^{\sqrt{9 - y^{2}}} {鈭�}_{0}^{3 - \sqrt{x^{2} + z^{2}}} f (x, y, z) d y d x d z$ represents the volume of the region is the right by the parabolic $(y - 3)^{2} = x^{2} + z^{2}$ bounded on the left by the xz plane.

Since from the given iterated integral we observe that $y = 3 - \sqrt{x^{2} + z^{2}}$

$\sqrt{x^{2} + z^{2}} = 3 - y 鈬� (3 - y)^{2} = x^{2} + z^{2}$

It represents the hyperbolic plane in xz

Thus the given iterated integral represents the volume of the region is the right by the parabolic $(y-3)^{2}=x^{2}+z^{2}$ bounded on the left by the $x z$ plane.

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

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. The three dimensional region.

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91影视

Describe the three-dimensional region expressed in each iterated integral in Exercises 35鈥�44.鈭�-33鈭�-9-z29-z2鈭�03-x2+z2f(x,y,z)dydxdz

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. The three dimensional region.

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

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Study anywhere. Anytime. Across all devices.

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