How to Integrate in Cylindrical 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 cylindrical and from spherical to cylindrical.
Below is a list of conversions from Cartesian to cylindrical.
Above is a diagram with point P{\displaystyle P} described in cylindrical coordinates. x=rcosθy=rsinθz=zr2=x2+y2{\displaystyle {\begin{aligned}x&=r\cos \theta \\y&=r\sin \theta \\z&=z\\r^{2}&=x^{2}+y^{2}\end{aligned}}} , 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 cylindrical coordinates. , dV=rdrdθdz{\displaystyle {\mathrm {d} }V=r{\mathrm {d} }r{\mathrm {d} }\theta {\mathrm {d} }z} 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. , Choose a coordinate system that allows for the easiest integration.
As with polar coordinates, the range of θ{\displaystyle \theta } is ,{\displaystyle ,} unless there are applications to integrating over more than the whole object. , Once everything is set up in cylindrical coordinates, simply integrate using any means possible and evaluate.
To save space in this article (and in your calculations) for the moment of inertia of a cone, it is useful to recognize the integral ∫02πsin2θdθ=π.{\displaystyle \int _{0}^{2\pi }\sin ^{2}\theta {\mathrm {d} }\theta =\pi .} -
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 cylindrical and from spherical to cylindrical.
Below is a list of conversions from Cartesian to cylindrical.
Above is a diagram with point P{\displaystyle P} described in cylindrical coordinates. x=rcosθy=rsinθz=zr2=x2+y2{\displaystyle {\begin{aligned}x&=r\cos \theta \\y&=r\sin \theta \\z&=z\\r^{2}&=x^{2}+y^{2}\end{aligned}}} , 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 cylindrical coordinates. , dV=rdrdθdz{\displaystyle {\mathrm {d} }V=r{\mathrm {d} }r{\mathrm {d} }\theta {\mathrm {d} }z} 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. , Choose a coordinate system that allows for the easiest integration.
As with polar coordinates, the range of θ{\displaystyle \theta } is ,{\displaystyle ,} unless there are applications to integrating over more than the whole object. , Once everything is set up in cylindrical coordinates, simply integrate using any means possible and evaluate.
To save space in this article (and in your calculations) for the moment of inertia of a cone, it is useful to recognize the integral ∫02πsin2θdθ=π.{\displaystyle \int _{0}^{2\pi }\sin ^{2}\theta {\mathrm {d} }\theta =\pi .}
About the Author
Peter Jenkins
Peter Jenkins is an experienced writer with over 17 years of expertise in consulting. Passionate about sharing practical knowledge, Peter creates easy-to-follow guides that help readers achieve their goals.
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