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math119: polar integral intro

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eggy 2023-02-07 11:19:15 -05:00
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@ -422,3 +422,13 @@ Functions can also be replaced to be bounded by the other if necessary.
&=\frac 1 4 x^4(2)\biggr|^2_0 \\ &=\frac 1 4 x^4(2)\biggr|^2_0 \\
&= 8 &= 8
\end{align*} \end{align*}
### Double polar integrals
The differential elements can be directly replaced:
$$dA=dxdy=\rho d\rho d\phi$$
In general, the radius should be the inner integral, and functions converted from Cartesian to polar forms.
$$\int^{\phi_2}_{\phi_1}\int^{\rho_2}_{\rho_1}f(\rho\cos\phi,\rho\sin\phi)\rho d\rho d\phi$$