Integration in Polar Coordinates

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In rectangular coordinates we find the area bounded by the curvethe x-axis, and the ordinates atandusing The corresponding problem in polar coordinates is that of determining the area bounded by the curveand the two radius vectorsandIn Fig. 4 this is the area bounded by the curve and the lines OA and OB.

We divide the-interval fromto up into n subintervals (not necessarily equal) having the magnitudesWe then draw the corresponding radius vectors, denoting their lengths byand draw the circular arcs as shown.

Remembering that the area of a circular sector having radius r and central angleiswe write down the following expression for the sum of the areas of the circular sectors:


The area bounded by the curve and the lines OA and OB is then equal to the limit of the following sum


where we are requiring that the largestas

Example: Compute the area bounded by the curve

The shaded area is three times the area of one leaf:

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