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Next: Substitution in multiple integrals. Up: Triple integrals. Previous: Triple integrals in cylindrical

Triple integrals in spherical coordinates.

  \fbox{
\begin{minipage}{10cm}
\begin{center}
Relations between cartesian coordin...
... = \sqrt{x^2+y^2+z^2} = \sqrt{r^2+z^2}$\end{tabular}\end{center}\end{minipage} }


  
Figure 14: The spherical coordinates of a point.
\begin{figure}
\mbox{\epsfig{file=SphericalCoord.eps,height=5cm} }
\end{figure}

Example 7.20   Let $\mathcal{C}$ be the cone whose equation is $z=\sqrt{x^2+y^2}$. By substitution from  2.6, we have:
\begin{align*}\rho \cos \phi & = \sqrt{ (r \cos \theta )^2 + ( r \sin \theta )^2...
...2 } \\
\rho \cos \phi &= \rho \sin \phi \\
\cos \phi &= \sin \phi
\end{align*}
i.e., an equation for $\mathcal{C}$ in cylindrical coordinates is the following:

\begin{displaymath}\phi = \frac {\pi}{4}.
\end{displaymath}


  
Figure 15: A cone.
\begin{figure}
\mbox{\epsfig{file=cone1.eps,height=4cm} }
\end{figure}

Example 7.21   See Fig.  16.


  
Figure 16: Simple surfaces in cylindrical coordinates.
\begin{figure}
\mbox{\subfigure[a sphere]{\epsfig{file=Sphere1.eps,height=4cm} }...
...ubfigure[a half-plane]{\epsfig{file=HalfPlane2.eps,height=4cm} }
}
\end{figure}

% latex2html id marker 10314
\fbox{
\begin{minipage}{9cm}
\begin{center}
Element...
...\Phi \; d \rho \; d \Phi \; d \theta
\end{equation*}\end{center}\end{minipage} }

Example 7.22   Compute the volume of the solid cut from the sphere whose equation is $\rho = 2$ by the cone $\theta = \frac {\pi}{4}$.


  
Figure 17: An ice-cream.
\begin{figure}
\mbox{\subfigure[in space]{\epsfig{file=IceCream.eps,height=4cm} ...
...uad
\subfigure[cut]{\epsfig{file=IceCream-cut.eps,height=4cm} }
}
\end{figure}

The domain of integration is given by the conditions:

\begin{displaymath}0 \leq \rho \leq 2 \qquad ; \qquad 0 \leq \Phi \leq \frac {\pi}{4} \qquad ; \qquad
0 \leq \theta \leq 2 \pi
\end{displaymath}

The volume is:
\begin{align*}V &= \int_0^{2 \pi} \int_0^{\pi / 4} \int_0^2 \rho^2 \; \sin \Phi ...
... 83 \right) \; d \theta \\
\quad &= \frac {8 \pi}{3} (\sqrt{2}+1).
\end{align*}


next up previous contents
Next: Substitution in multiple integrals. Up: Triple integrals. Previous: Triple integrals in cylindrical
Noah Dana-Picard
2001-05-30