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Use spherical coordinates. Evaluate /| y² dV, where E is the solid hemisphere x2 + y² + z² s 9, y 2 0.

Use spherical coordinates. Evaluate /| y² dV, where E is the solid hemisphere x2 + y² + z² s 9, y 2 0.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Use Spherical Coordinates** **Evaluate:** \[ \iiint\limits_E y^2 \, dV \] where \( E \) is the solid hemisphere defined by \( x^2 + y^2 + z^2 \leq 9 \), \( y \geq 0 \). **Explanation:** The problem involves evaluating a triple integral of \( y^2 \) over a hemisphere using spherical coordinates. The hemisphere is represented by the inequality \( x^2 + y^2 + z^2 \leq 9 \), which describes a sphere with radius 3, constrained to the half-space where \( y \geq 0 \). By converting to spherical coordinates, the boundaries and variable \( y \) need to be expressed in terms of the spherical variables: radius \(\rho\), inclination \(\theta\), and azimuthal angle \(\phi\). This facilitates integration within the given constraints. The integral setup in spherical coordinates would generally involve transforming the integrand and limits accordingly, utilizing the spherical coordinate relationships: - \( x = \rho \sin \phi \cos \theta \) - \( y = \rho \sin \phi \sin \theta \) - \( z = \rho \cos \phi \) The differential volume element is \( dV = \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta \).
Transcribed Image Text:**Use Spherical Coordinates** **Evaluate:** \[ \iiint\limits_E y^2 \, dV \] where \( E \) is the solid hemisphere defined by \( x^2 + y^2 + z^2 \leq 9 \), \( y \geq 0 \). **Explanation:** The problem involves evaluating a triple integral of \( y^2 \) over a hemisphere using spherical coordinates. The hemisphere is represented by the inequality \( x^2 + y^2 + z^2 \leq 9 \), which describes a sphere with radius 3, constrained to the half-space where \( y \geq 0 \). By converting to spherical coordinates, the boundaries and variable \( y \) need to be expressed in terms of the spherical variables: radius \(\rho\), inclination \(\theta\), and azimuthal angle \(\phi\). This facilitates integration within the given constraints. The integral setup in spherical coordinates would generally involve transforming the integrand and limits accordingly, utilizing the spherical coordinate relationships: - \( x = \rho \sin \phi \cos \theta \) - \( y = \rho \sin \phi \sin \theta \) - \( z = \rho \cos \phi \) The differential volume element is \( dV = \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta \).
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