Mathematics: A region R is shown. Decide whether to use polar coordinates or rectangular coordinates and write

A region R is shown. Decide whether to use polar

breannamartinez9486

28.01.2022

Writing the integral in polar coordinates is less taxing:

$\displaystyle\iint_Rf(x,y)\,\mathrm dA=\iiint_Rrf(r\cos\theta,r\sin\theta)\,\mathrm dr\,\mathrm d\theta$

The region R is captured by the set of points (in polar, of course)

$R=\left\{(r,\theta)\mid1\le r\le3\land\dfrac{5\pi}4\le\theta\le2\pi\right\}$

so that the integral is

$\displaystyle\int_{5\pi/4}^{2\pi}\int_1^3rf(r\cos\theta,r\sin\theta)\,\mathrm dr\,\mathrm d\theta$

In rectangular coordinates, we have to split up the region so that

$R=R_1\cup R_2\cup R_3$

where

$R_1=\left\{(x,y)\mid-\dfrac3{\sqrt2}\le x\le-\dfrac1{\sqrt2}\land-\sqrt{9-x^2}\le y\le x\right\}$

$R_2=\left\{(x,y)\mid-\dfrac1{\sqrt2}\le x\le1\land-\sqrt{9-x^2}\le y\le -\sqrt{1-x^2}\right\}$

$R_3=\left\{(x,y)\mid1\le x\le3\land-\sqrt{9-x^2}\le y\le0\right\}$

and the integral is split into 3 over these regions,

$\displaystyle\iint_Rf(x,y)\,\mathrm dA=\sum_{n=1}^3\iint_{R_n}f(x,y)\,\mathrm dx\,\mathrm dy$

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