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'example of integration with respect to surface area of the paraboloid'
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| Title of object: |
example of integration with respect to surface area of the paraboloid |
| Canonical Name: |
Example7OfIntegrationWithRespectToSurfaceArea |
| Type: |
Example |
| Created on: |
2005-01-27 16:35:41 |
| Modified on: |
2005-10-15 15:39:23 |
| Classification: |
msc:28A75 |
Revision comment (for changes between this and next version):
| fixed, so accept correction |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
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Content:
\PMlinkescapeword{links}
\PMlinkescapeword{quick}
In this example, we examine the paraboloid given by the equation $z = x^2 + 3 y^2$. We have
$$\sqrt{1 + \left( \frac{\partial g}{\partial x} \right)^2 + \left( \frac{\partial g}{\partial y} \right)^2} =$$
$$\sqrt{1 + \left( 2 x \right)^2 + \left( 6 y \right)^2} = \sqrt{ 4 x^2 + 36 y^2 }$$
and, hence
$$\int_S f(x,y) \, d^2 A = \int f(x,y) \sqrt{ 4 x^2 + 36 y^2 } \, dx \, dy.$$
\PMlinkescapetext{\sl Quick links:}
\begin{itemize}
\item \PMlinkid{main entry}{6660}
\item \PMlinkid{previous example}{6669}
\item {next example}
\end{itemize} |
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