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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Triple Integrals in Cylindrical and Spherical Coordinates">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
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\section{Triple\ Integrals In Cylindrical and Spherical Coordinates\label%
{Triple Integrals in Cylindrical and Spherical Coordinates}}

\subsubsection{Triple Integration In Cylindrical Coordinates\label%
{Formula for Triple Integration in Cylindrical Coordinates}}

Suppose that $f$ is continuous and 
\[
E=\left\{ (x,y,z)\ |\ (x,y)\in D,\ \phi _{1}(x,y)\leq z\leq \phi
_{2}(x,y)\right\} 
\]
where$\;D$ is given in polar coordinates by 
\[
D=\left\{ (r,\theta )\ |\ \alpha \leq \theta \leq \beta \text{,}\
h_{1}\left( \theta \right) \leq r\leq h_{2}\left( \theta \right) \right\} 
\]
The triple integral of $f$ over $E$ in cylindrical coordinates is 
\[
\iiint\limits_{E}f(x,y,z)dV=\int_{\alpha }^{\beta }\int_{h_{1}\left( \theta
\right) }^{h_{2}\left( \theta \right) }\int_{\phi _{1}(r\cos \theta ,r\sin
\theta )}^{\phi _{2}(r\cos \theta ,r\sin \theta )}f(r\cos \theta ,r\sin
\theta )rdzdrd\theta 
\]

\subsubsection{Triple Integration In Spherical Coordinates\label%
{Formula for Triple Integration in Spherical Coordinates}}

Suppose that $f$ is continuous and $E$ is a spherical wedge given by 
\[
E=\left\{ (\rho ,\theta ,\phi )\ |\ a\leq \rho \leq b\text{,}\ \alpha \leq
\theta \leq \beta \text{,}\ c\leq \phi \leq d\right\} 
\]
The triple integral of $f$ over $E$ in spherical coordinates is 
\[
\iiint\limits_{E}f(x,y,z)dV=\int_{c}^{d}\int_{\alpha }^{\beta
}\int_{a}^{b}f(\rho \sin \phi \cos \theta ,\rho \sin \phi \sin \theta ,\rho
\cos \phi )\rho ^{2}\sin \phi d\rho d\theta d\phi 
\]

\subsection{Related Reference Topics}

{\small \hyperref{Cylindrical coordinates in calculus}{}{}{%
Cal10_10.tex#Cylindrical and Spherical Coordinates}}

{\small \hyperref{Cylindrical coordinates in geometry}{}{}{%
GE06_01.tex#Cylindrical coordinates}}

{\small \hyperref{Relations between rectangular, cylindrical, and spherical
coordinates}{}{}{GE06_01.tex#Relations between coordinate systems}}

{\small \hyperref{Spherical coordinates in calculus}{}{}{%
Cal10_10.tex#Cylindrical and Spherical Coordinates}}

{\small \hyperref{Spherical coordinates in geometry}{}{}{%
GE06_01.tex#Spherical coordinates}}

\subsection{Related Scientific Notebook Topics}

\vspace{1pt}{\small \hyperref{Iterated integrals}{}{}{%
../../Help/DM7-8.tex#Iterated integrals}}

{\small \hyperref{Order of integration}{}{}{../../Help/DM7-8.tex#Order of
integration}}

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