3 double integrals in polar coordinates we will see

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Unformatted text preview: We will see that polar coordinates is a useful tool for evaluating some double integrals, especially the regions that have some type of radial symmetry. Instead of chopping the region up into rectangular subregions, we will use rays emanating from the origin and concentric circular arcs as shown below: We will derive the area of this new subregion: In rectangular coordinates we had our ray that chopped up the region R being either horizontal or vertical. This is how we found our outside limits. Then we found the curves the ray started at and ended at to get our inside limits. This idea is similar in polar except our ray is now always emanating from the origin and sweeps through an angle. The angle we start at and end at in the sweep determine our outside limits. The inside limits are determined as always, from where the ray hits the first curve to where the ray hits the second curve. (these curves are functions of theta ie: ) The key to using polar is to understand the above and remembering Set up the double integral that represents the area of the region bounded by the curves: Outside the circle and inside the lemniscate . Find where the two curves intersect. Figure...
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