| Version 3 |
Version 2 |
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In this example, we examine the paraboloid given by the equation $z = x^2 + 3 y^2$. We have
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In this example, we examine the paraboloid given by the equation $z = x^2 + 3 y^3$. We have
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| $$\sqrt{1 + \left( \frac{\partial g}{\partial x} \right)^2 + \left( \frac{\partial g}{\partial y} \right)^2} =$$ |
$$\sqrt{1 + \left( \frac{\partial g}{\partial x} \right)^2 + \left( \frac{\partial g}{\partial y} \right)^2} =$$ |
| $$\sqrt{1 + \left( 2 x \right)^2 + \left( 6 y \right)^2} = \sqrt{ 4 x^2 + 36 y^2 }$$ |
$$\sqrt{1 + \left( 2 x \right)^2 + \left( 6 y \right)^2} = \sqrt{ 4 x^2 + 36 y^2 }$$ |
| and, hence |
and, hence |
| $$\int_S f(x,y) \, d^2 A = \int f(x,y) \sqrt{ 4 x^2 + 36 y^2 } \, dx \, dy.$$ |
$$\int_S f(x,y) \, d^2 A = \int f(x,y) \sqrt{ 4 x^2 + 36 y^2 } \, dx \, dy.$$ |
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| {\sl Quick links:} |
{\sl Quick links:} |
| \begin{itemize} |
\begin{itemize} |
| \item \PMlinkid{main entry}{6660} |
\item \PMlinkid{main entry}{6660} |
| \item \PMlinkid{previous example}{6669} |
\item \PMlinkid{previous example}{6669} |
| \item {next example} |
\item {next example} |
| \end{itemize} |
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