How to Integrate in Spherical Coordinates
Recall the coordinate conversions., Set up the coordinate-independent integral., Set up the volume element., Set up the boundaries., Integrate.
Step-by-Step Guide
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Step 1: Recall the coordinate conversions.
Coordinate conversions exist from Cartesian to spherical and from cylindrical to spherical.
Below is a list of conversions from Cartesian to spherical.
Above is a diagram with point P{\displaystyle P} described in spherical coordinates. x=ρsinϕcosθy=ρsinϕsinθz=ρcosϕρ2=x2+y2+z2{\displaystyle {\begin{aligned}x&=\rho \sin \phi \cos \theta \\y&=\rho \sin \phi \sin \theta \\z&=\rho \cos \phi \\\rho ^{2}&=x^{2}+y^{2}+z^{2}\end{aligned}}} In the example where we calculate the moment of inertia of a ball, x2+y2=ρ2sin2ϕ{\displaystyle x^{2}+y^{2}=\rho ^{2}\sin ^{2}\phi } will be useful.
Make sure you know why this is the case. , We are dealing with volume integrals in three dimensions, so we will use a volume differential dV{\displaystyle {\mathrm {d} }V} and integrate over a volume V.{\displaystyle V.} ∫VdV{\displaystyle \int _{V}{\mathrm {d} }V} Most of the time, you will have an expression in the integrand.
If so, make sure that it is in spherical coordinates. , dV=ρ2sinϕdρdϕdθ{\displaystyle {\mathrm {d} }V=\rho ^{2}\sin \phi {\mathrm {d} }\rho {\mathrm {d} }\phi {\mathrm {d} }\theta } Those familiar with polar coordinates will understand that the area element dA=rdrdθ.{\displaystyle {\mathrm {d} }A=r{\mathrm {d} }r{\mathrm {d} }\theta .} This extra r stems from the fact that the side of the differential polar rectangle facing the angle has a side length of rdθ{\displaystyle r{\mathrm {d} }\theta } to scale to units of distance.
A similar thing is occurring here in spherical coordinates. , Choose a coordinate system that allows for the easiest integration.
Notice that ϕ{\displaystyle \phi } has a range of ,{\displaystyle ,} not .{\displaystyle .} This is because θ{\displaystyle \theta } already has a range of ,{\displaystyle ,} so the range of ϕ{\displaystyle \phi } ensures that we don’t integrate over a volume twice. , Once everything is set up in spherical coordinates, simply integrate using any means possible and evaluate. -
Step 2: Set up the coordinate-independent integral.
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Step 3: Set up the volume element.
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Step 4: Set up the boundaries.
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Step 5: Integrate.
Detailed Guide
Coordinate conversions exist from Cartesian to spherical and from cylindrical to spherical.
Below is a list of conversions from Cartesian to spherical.
Above is a diagram with point P{\displaystyle P} described in spherical coordinates. x=ρsinϕcosθy=ρsinϕsinθz=ρcosϕρ2=x2+y2+z2{\displaystyle {\begin{aligned}x&=\rho \sin \phi \cos \theta \\y&=\rho \sin \phi \sin \theta \\z&=\rho \cos \phi \\\rho ^{2}&=x^{2}+y^{2}+z^{2}\end{aligned}}} In the example where we calculate the moment of inertia of a ball, x2+y2=ρ2sin2ϕ{\displaystyle x^{2}+y^{2}=\rho ^{2}\sin ^{2}\phi } will be useful.
Make sure you know why this is the case. , We are dealing with volume integrals in three dimensions, so we will use a volume differential dV{\displaystyle {\mathrm {d} }V} and integrate over a volume V.{\displaystyle V.} ∫VdV{\displaystyle \int _{V}{\mathrm {d} }V} Most of the time, you will have an expression in the integrand.
If so, make sure that it is in spherical coordinates. , dV=ρ2sinϕdρdϕdθ{\displaystyle {\mathrm {d} }V=\rho ^{2}\sin \phi {\mathrm {d} }\rho {\mathrm {d} }\phi {\mathrm {d} }\theta } Those familiar with polar coordinates will understand that the area element dA=rdrdθ.{\displaystyle {\mathrm {d} }A=r{\mathrm {d} }r{\mathrm {d} }\theta .} This extra r stems from the fact that the side of the differential polar rectangle facing the angle has a side length of rdθ{\displaystyle r{\mathrm {d} }\theta } to scale to units of distance.
A similar thing is occurring here in spherical coordinates. , Choose a coordinate system that allows for the easiest integration.
Notice that ϕ{\displaystyle \phi } has a range of ,{\displaystyle ,} not .{\displaystyle .} This is because θ{\displaystyle \theta } already has a range of ,{\displaystyle ,} so the range of ϕ{\displaystyle \phi } ensures that we don’t integrate over a volume twice. , Once everything is set up in spherical coordinates, simply integrate using any means possible and evaluate.
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Patrick Hughes
Patrick Hughes has dedicated 4 years to mastering education and learning. As a content creator, Patrick focuses on providing actionable tips and step-by-step guides.
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