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Surface area.

    

% latex2html id marker 10826
\fbox{
\begin{minipage}{10cm}
\par \begin{center}
\...
...}f \cdot \overrightarrow{p} \end{vmatrix}} \; dA
\end{equation*}\end{minipage} }

Example 8.25   We compute the area of the surface cut from the paraboloid z=x2+y2 by the plane z=4.
  
Figure 6:
\begin{figure}
\mbox{\epsfig{file=ParaboloidArea.eps,height=5cm} }
\end{figure}

We have:
\begin{align*}\text{Area} & =\iint_{\mathcal{R}} \frac {\begin{vmatrix}\overrigh...
... dA \\
\quad &= \iint_{x^2+y^2=4} \sqrt{4x^2+4y^2+1} \; dx \; dy .
\end{align*}
We use polar coordinates; r2=x2+y2, thus we have:
\begin{align*}\text{Area} & = \int_0^{2 \pi} \int_0^2 \sqrt{r^2+1} \; r \; dr \;...
...7^{3/2} -1 )\; d \theta \\
\quad &= \frac {\pi}{6} (17^{3/2} -1 ).
\end{align*}

Example 8.26  



Noah Dana-Picard
2001-05-30