4.3. Please solve the following exercise. Exercise 3Use cylindrical coordinates to find the volume of the solid bounded by sphere x² + y² + z² =2and the inside of the cylinder Xa22+y²=a2Plot the volume.

Given Information:Sphere: Cylinder: To find:The volume of the solid bounded by the sphere and inside…

Answered: Use cylindrical coordinates to find the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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4.3. Please solve the following exercise.

Exercise 3
Use cylindrical coordinates to find the volume of the solid bounded by sphere x² + y² + z² =
2
and the inside of the cylinder X
a
2
2
+y²
=
a
2
Plot the volume.

Expert Solution

Step 1: Analysis and Introduction

Given Information:

Sphere: $x squared plus y squared plus z squared equals a squared$

Cylinder: $open parentheses x minus a over 2 close parentheses squared plus y squared equals open parentheses a over 2 close parentheses squared$

To find:

The volume of the solid bounded by the sphere and inside the cylinder.

Concept used:

The cartesian coordinate $open parentheses x comma y comma z close parentheses$ is converted to cylindrical coordinate $open parentheses r comma theta comma z close parentheses$ by assuming $x equals r cos open parentheses theta close parentheses comma space y equals r sin open parentheses theta close parentheses$ and $theta$ as usual polar angle taken by the region of the solid.

Formula used:

$integral x to the power of n d x equals fraction numerator x to the power of n plus 1 end exponent over denominator n plus 1 end fraction plus c$ , $for all n element of straight real numbers minus open curly brackets negative 1 close curly brackets$ . Here, $c$ is an integrating constant.

$fraction numerator d over denominator d x end fraction open parentheses sin open parentheses x close parentheses close parentheses equals cos open parentheses x close parentheses$

$fraction numerator d over denominator d x end fraction open parentheses cos open parentheses x close parentheses close parentheses equals negative sin open parentheses x close parentheses$

Algebraic Identity: $open parentheses a minus b close parentheses squared equals a squared plus b squared minus 2 a b$