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Section 18.3 Double integrals over polar regions
Objectives
-
Use the change-of-variables formula to rewrite double integrals in new coordinates.
-
Apply polar coordinates to compute quantities such as moments of inertia.
-
Derive the area under the Gaussian curve using polar methods.
-
Change of variables in an integral:
\begin{equation*}
\iint_{R} f(x,y)\dd{x}\dd{y}=\iint_{S} f(x(u,v),y(u,v))\abs{\pdv{(x,y)}{(u,v)}}\dd{u}\dd{v}
\end{equation*}
-
Polar Jacobian:
\begin{equation*}
\abs{\pdv{(x,y)}{(r,\theta)}} = r\text{.}
\end{equation*}
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Simplified moments of inertia for anything involving circular symmetry
-
Do the Gaussian integral using the polar trick
