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Next: Triple integrals in spherical Up: Triple integrals. Previous: Applications: masses, moments, center

   
Triple integrals in cylindrical coordinates.

  \fbox{
\begin{minipage}{9cm}
\begin{center}
Relations between cartesian coordina...
...x^2+y^2 \; ; \; \tan \theta = \frac yx$\end{tabular}\end{center}\end{minipage} }


  \begin{figure}
\mbox{\subfigure[The cylindrical coordinates of a point.]{
\epsf...
...entary area.]
{\epsfig{file=ElementAreaCylind.eps,height=5cm} }
}
\end{figure}

Example 7.16   Let $\mathcal{C}$ be the cylinder whose equation is (x-1)2+y2=4. By substitution from  2.5, we have:

\begin{displaymath}( r \cos \theta -1)^2 + ( r \sin \theta )^2 =4
\end{displaymath}

i.e., an equation for $\mathcal{C}$ in cylindrical coordinates is the following:

\begin{displaymath}r^2 -2r \cos \theta -3 = 0.
\end{displaymath}

Example 7.17   Let $\mathcal{C}$ be the cylinder whose equation is (x-2)2+y2=4. By substitution from  2.5, we have:

\begin{displaymath}( r \cos \theta -2)^2 + ( r \sin \theta )^2 =4
\end{displaymath}

i.e., an equation for $\mathcal{C}$ in cylindrical coordinates is the following:

\begin{displaymath}r = 2 \cos \theta.
\end{displaymath}


  
Figure 11: A cylinder.
\begin{figure}
\mbox{\epsfig{file=cylindre2.eps,height=4cm} }
\end{figure}

% latex2html id marker 10250
\fbox{
\begin{minipage}{9cm}
\begin{center}
Element...
...tion*}
dV= dz \; r \; dr \; d \theta
\end{equation*}\end{center}\end{minipage} }

Example 7.18   Compute the volume of the domain enclosed by the xy-plane, the cylinder whose equation is x2+y2=1 and the paraboloid whose equation is z=4-x2-y2.


  
Figure 12: A cylinder and a paraboloid.
\begin{figure}
\mbox{\epsfig{file=ParabCyl.eps,height=5cm} }
\end{figure}

The coordinates inequalities for the given domain are:

\begin{displaymath}0 \leq r \leq 1 \qquad ; \qquad 0 \leq \theta \leq 2 \pi \qquad ; \qquad 0 \leq z \leq 4-r^2
\end{displaymath}

We have:
\begin{align*}V &= \underset{\mathcal{D}}{\int \int \int} dV
= \int_0^{2 \pi} \...
...0^{2 \pi} \frac {11}{3} \; d \theta \\
\quad &= \frac {22 \pi}{3}.
\end{align*}

Example 7.19   Compute the volume of the domain enclosed by the paraboloids P1 and P2 whose respective equations are z=x2+y2 and z=4-x2-y2.


  
Figure 13: Two paraboloids.
\begin{figure}
\mbox{\subfigure[in space]{\epsfig{file=TwoParaboloids.eps,height...
...e[$rz-$ plane]{\epsfig{file=TwoParaboloids-cut.eps,height=5cm} }
}
\end{figure}

The coordinates inequalities for the given domain are:

\begin{displaymath}0 \leq r \leq 2 \qquad ; \qquad 0 \leq \theta \leq 2 \pi \qquad ; \qquad r^2 \leq z \leq 4-r^2
\end{displaymath}

The two paraboloids intersect for z=2, i.e. $r= \sqrt{2}$. We have:
\begin{align*}V &= \int_0^{2 \pi} \int_0^{\sqrt{2}} \int_{r^2}^{4-r^2} \; dz \; ...
...{2}} \; d \theta \\
\quad &= \int_0^{2 \pi} 2 \; d \theta = 4 \pi.
\end{align*}


next up previous contents
Next: Triple integrals in spherical Up: Triple integrals. Previous: Applications: masses, moments, center
Noah Dana-Picard
2001-05-30