Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

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

The three-dimensional region is given by the planer equation,

$x^{2} + y^{2} + z^{2} = 3^{2}$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information

We are given,

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

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

The given iterated integral $\int_{- 3}^{3} \int_{\sqrt{9 - x^{2}}}^{\sqrt{9 - x^{2}}} \int_{- \sqrt{9 - x^{2} - y^{2}}}^{\sqrt{9 - x^{2} - y^{2}}} f (x, y, z) d z d y d x$ represents the volume of the sphere with radius $3$ and centered at origin.

Since from the integral limits we observe that localid="1650347847843" $z = \sqrt{9 - x^{2} - y^{2}}$ to $z = \sqrt{9 - x^{2} - y^{2}}$

Squaring on both sides,

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

Simplifying and rearranging, we get

$x^{2} + y^{2} + z^{2} = 3^{2}$

This represents the equation of sphere with radius $3$ and centered at origin

Hence the given iterated integral represents the volume of the sphere with radius $3$ and centered at origin.

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Calculus

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Short Answer

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

Step by Step Solution

Step 1. Given Information

Step 2. The three dimensional region.

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Calculus

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Describe the three-dimensional region expressed in each iterated integral in Exercises 35–44.∫-33∫-9-x29-x2∫-9-x2-y29-x2-y2f(x,y,z)dzdydx

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