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https://texercises.com/exercise/volume-integral-and-spherical-symmetry/
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Exercise:
Show that the volume egral for a function fr with spherical symmetry is given by _V fr textrmdV pi_^infty fr r^ textrmdr

Solution:
We use the convention in the figure for the shperical coordinates: center includegraphicswidthtextwidth#image_path:spherical-coordinates-# center The volume egral for a general function frtheta phi in spherical coordinates is given by _V frthetaphi textrmdV _r^infty_theta^pi _phi^pifrthetaphi r^ sinthetatextrmdrtextrmdthetatextrmdphi A function fr with spherical symmetry can be considered to be a constant with respect to the egration with respect to the angles theta and phi. Thus we find _V fr textrmdV _^infty fr textrmdr _theta^pi _phi^pisintheta textrmdtheta textrmdphi _^infty fr textrmdr pi _theta^pi sintheta textrmdtheta pi _r^infty fr textrmdr left-costheta rightBig|_^pi pi _r^infty fr textrmdr left-cospi+cosright pi _r^infty fr textrmdr pi _^infty fr textrmdr
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Exercise:
Show that the volume egral for a function fr with spherical symmetry is given by _V fr textrmdV pi_^infty fr r^ textrmdr

Solution:
We use the convention in the figure for the shperical coordinates: center includegraphicswidthtextwidth#image_path:spherical-coordinates-# center The volume egral for a general function frtheta phi in spherical coordinates is given by _V frthetaphi textrmdV _r^infty_theta^pi _phi^pifrthetaphi r^ sinthetatextrmdrtextrmdthetatextrmdphi A function fr with spherical symmetry can be considered to be a constant with respect to the egration with respect to the angles theta and phi. Thus we find _V fr textrmdV _^infty fr textrmdr _theta^pi _phi^pisintheta textrmdtheta textrmdphi _^infty fr textrmdr pi _theta^pi sintheta textrmdtheta pi _r^infty fr textrmdr left-costheta rightBig|_^pi pi _r^infty fr textrmdr left-cospi+cosright pi _r^infty fr textrmdr pi _^infty fr textrmdr
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Attributes & Decorations
Branches
Integrals
Tags
spherical coordinates, volume
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Difficulty
(1, default)
Points
0 (default)
Language
ENG (English)
Type
Calculative / Quantity
Creator by
Decoration