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II. 2 + y?) dv. JE

II. 2 + y?) dv. JE

Advanced Engineering Mathematics
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.
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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