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6. Use spherical coordinates to evaluate the triple integral of the function f(x, y, z)= x over the solid, bounded by the surfaces x² + y² +=² ≤1; x,y,z ≤0

6. Use spherical coordinates to evaluate the triple integral of the function f(x, y, z)= x over the solid, bounded by the surfaces x² + y² +=² ≤1; x,y,z ≤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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### Problem Statement **Use spherical coordinates to evaluate the triple integral of the function \( f(x, y, z) = x \) over the solid bounded by the surfaces \( x^2 + y^2 + z^2 \leq 1 \) and \( x, y, z \leq 0 \).** This problem requires you to convert the given integral into spherical coordinates and then evaluate the integral over the specified solid region. #### Solution Steps: 1. **Convert Cartesian coordinates to spherical coordinates:** - \( x = \rho \sin \phi \cos \theta \) - \( y = \rho \sin \phi \sin \theta \) - \( z = \rho \cos \phi \) - Jacobian for spherical coordinates: \( \rho^2 \sin \phi \) 2. **Set the bounds of integration:** - The solid is a portion of a sphere. In spherical coordinates, the surface \( x^2 + y^2 + z^2 = 1 \) corresponds to \( \rho = 1 \). - The bounds for \( \rho \) are from 0 to 1. - The inequality \( x, y, z \leq 0 \) means that \( \theta \) ranges from \( \pi/2 \) to \( \pi \) and \( \phi \) ranges from \( \pi/2 \) to \( \pi \). 3. **Set up the integral using spherical coordinates:** \[ \iiint_V x \, dV = \int_{\pi/2}^{\pi} \int_{\pi/2}^{\pi} \int_0^1 (\rho \sin \phi \cos \theta) \cdot (\rho^2 \sin \phi) \, d\rho \, d\phi \, d\theta \] 4. **Simplify the integrand:** \[ \rho^3 \sin^2 \phi \cos \theta \] 5. **Evaluate the integral:** \[ \iiint_V x \, dV = \int_{\pi/2}^{\pi} \int_{\pi/2}^{\pi} \int_0^1 \rho^3 \sin^2 \phi \cos \theta \, d\rho \
Transcribed Image Text:### Problem Statement **Use spherical coordinates to evaluate the triple integral of the function \( f(x, y, z) = x \) over the solid bounded by the surfaces \( x^2 + y^2 + z^2 \leq 1 \) and \( x, y, z \leq 0 \).** This problem requires you to convert the given integral into spherical coordinates and then evaluate the integral over the specified solid region. #### Solution Steps: 1. **Convert Cartesian coordinates to spherical coordinates:** - \( x = \rho \sin \phi \cos \theta \) - \( y = \rho \sin \phi \sin \theta \) - \( z = \rho \cos \phi \) - Jacobian for spherical coordinates: \( \rho^2 \sin \phi \) 2. **Set the bounds of integration:** - The solid is a portion of a sphere. In spherical coordinates, the surface \( x^2 + y^2 + z^2 = 1 \) corresponds to \( \rho = 1 \). - The bounds for \( \rho \) are from 0 to 1. - The inequality \( x, y, z \leq 0 \) means that \( \theta \) ranges from \( \pi/2 \) to \( \pi \) and \( \phi \) ranges from \( \pi/2 \) to \( \pi \). 3. **Set up the integral using spherical coordinates:** \[ \iiint_V x \, dV = \int_{\pi/2}^{\pi} \int_{\pi/2}^{\pi} \int_0^1 (\rho \sin \phi \cos \theta) \cdot (\rho^2 \sin \phi) \, d\rho \, d\phi \, d\theta \] 4. **Simplify the integrand:** \[ \rho^3 \sin^2 \phi \cos \theta \] 5. **Evaluate the integral:** \[ \iiint_V x \, dV = \int_{\pi/2}^{\pi} \int_{\pi/2}^{\pi} \int_0^1 \rho^3 \sin^2 \phi \cos \theta \, d\rho \
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