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Next: Triple integrals. Up: Double integrals. Previous: Applications: areas, moments, center

Double integrals in polar 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 polar coordinates of a point.]
{\epsfig{fil...
...mentary area.]
{\epsfig{file=ElementAreaPolar.eps,height=5cm} }
}
\end{figure}

% latex2html id marker 10094
\fbox{
\begin{minipage}{9cm}
\begin{center}
Element...
...n{equation*}
dA= r \; dr \; d \theta
\end{equation*}\end{center}\end{minipage} }

% latex2html id marker 10098
\fbox{
\begin{minipage}{9cm}
\begin{center}
Area of...
...nt_{\mathcal{R}} r \; dr \; d \theta
\end{equation*}\end{center}\end{minipage} }

Example 7.8   We begin with a well-known example: compute the area of the unit disk $\mathcal{D}= \{ (x,y) \in \mathbb{R} ^2 \; ; \; x^2+y^2 =1 \}$.


  
Figure 5: The unit disk.
\begin{figure}
\mbox{\epsfig{file=unitdisk-polar.eps,height=5cm} }
\end{figure}

We have:


\begin{align*}S_{\mathcal{D}} &= \int \int _{\mathcal{D}} dA = \int_0^{2 \pi } \...
...2 \pi } \frac 12 d \theta \\
\quad & = 2 \pi \cdot \frac 12 = \pi.
\end{align*}

Example 7.9   Find the center of mass of the ``banana'' enclosed by the unit circle and the cardioid defined by the equation $r= 1+ \cos \theta$, assuming that the density is equal to 1 everywhere.

The ``banana'' has been drawn with MuPaD (http://www.sciface.com).


  
Figure 6: The unit disk and a cardioid.
\begin{figure}
\mbox{\epsfig{file=banana.eps,height=10cm} }
\end{figure}


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