II. 2 + y?) dv. JE
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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![**Use Spherical Coordinates**
Evaluate
\[
\iiint_E (x^2 + y^2) \, dV
\]
where \( E \) lies between the spheres \( x^2 + y^2 + z^2 = 1 \) and \( x^2 + y^2 + z^2 = 4 \).
---
This problem involves integrating the function \( x^2 + y^2 \) over the region \( E \), which is the volume between two concentric spheres. The inner sphere has a radius of 1, while the outer sphere has a radius of 2. The task is facilitated by transforming the integral into spherical coordinates, where \( x = \rho \sin \phi \cos \theta \), \( y = \rho \sin \phi \sin \theta \), and \( z = \rho \cos \phi \).
Here, \( \rho \) ranges from 1 to 2, \( \phi \) ranges from 0 to \(\pi\), and \( \theta \) ranges from 0 to \(2\pi\). The integral must be evaluated accordingly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5abdcc13-7dd6-4cb8-8b78-277f334acbdf%2F94d8cd05-3a51-424b-9bf0-3d84a943a6c8%2Fszrdhdk_processed.png&w=3840&q=75)
Transcribed Image Text:**Use Spherical Coordinates**
Evaluate
\[
\iiint_E (x^2 + y^2) \, dV
\]
where \( E \) lies between the spheres \( x^2 + y^2 + z^2 = 1 \) and \( x^2 + y^2 + z^2 = 4 \).
---
This problem involves integrating the function \( x^2 + y^2 \) over the region \( E \), which is the volume between two concentric spheres. The inner sphere has a radius of 1, while the outer sphere has a radius of 2. The task is facilitated by transforming the integral into spherical coordinates, where \( x = \rho \sin \phi \cos \theta \), \( y = \rho \sin \phi \sin \theta \), and \( z = \rho \cos \phi \).
Here, \( \rho \) ranges from 1 to 2, \( \phi \) ranges from 0 to \(\pi\), and \( \theta \) ranges from 0 to \(2\pi\). The integral must be evaluated accordingly.
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