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Integration in Polar Coordinates

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:

ole16.gif

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

ole17.gif

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: